Wales’ oldest club escaped a winding up petition earlier this month, but has been forced to postpone two league matches.
Neath once dominated the amateur game in Wales in the late 1980s, but are currently bottom of the semi-professional Principality Premiership.
Fans have called for Cuddy to leave while 68 former players, including ex-Wales internationals Paul Thorburn, Duncan Jones and Rowland Phillips, have put their name to a statement organised by Martyn Morris posted on the club’s website urging ownership to change hands.
Cuddy initially indicated he wanted to stay and build up the club again, but now says he is prepared to leave the club because of his ill-health and believes a takeover could be completed “within weeks”.
Cuddy has been diagnosed with Neurosarcoidosis, which can have stroke-like symptoms, that has affected his mobility and speech.
“I don’t want to continue being directly involved with Neath Rugby club,” Cuddy told BBC Sport Wales.
“I want to hand it on to people who are fitter and healthier and able to run the club as it deserves to be run.
“I have been involved with Neath for almost 30 years and am conscious of where the club is now and it can’t be healthy anymore under my stewardship.
“There have been some family pressures while doctor’s advice has told me I can’t bear this as I have been doing.
“If I don’t listen to the doctor I am a fool and I have never been a fool.
“In my heart I would like to continue and want to battle it. Two years ago I would have.
“As I am now, with my health not being good, I would be putting my life at risk if I continued to battle this because my progress has stalled under the stress of the situation.”
Neath postponed two matches, against Bedwas and RGC 1404, after being unable to field sides.
Players and coaching staff have left the club because the perilous financial position.
Club accounts were frozen, which meant wages could not be paid and other financial commitments could not be met.
In July 2018 Cuddy’s construction firm went into administration which he blames on his poor health.
“The difficulty stemmed from my illness,” said Cuddy.
“I have been ill for two years and over that time the business suffered.
“I built the business, but in common with a lot of family companies I don’t think there was the infrastructure for it to cope without me.
“It has been a difficult two years. The effect of the business crystallised quickly and my illness has hurt the rugby club.
“I and the team running the club at that time realised how critical this year was and signed contracts at a time the club needed new players and blood because Neath has been at the bottom of the table for a number of years.
“In an effort to keep the club functioning and, not understanding the critical position Cuddy Group was in at that time, contracts were signed at a time when the business failed.
“The amount of support I was putting into the club could not be sustained. That is where the problems came from.”
Cuddy has previously said Neath were not helped by an under-performing expensive squad which lost the opening 10 games of the 2018/19 season.
Financial troubles culminated in a court hearing when finance company Jardine Norton brought a petition against the club.
A judge found Neath RFC was insolvent, but could not wind it up because paperwork relating to a £31,000 debt was “unclear”.
During the hearing it also emerged the club owes Her Majesty’s Revenue and Customs (HMRC) just over £10,000, and Neath Port Talbot council may also be owed money.
“The Jardine Norton debt was taken on to help support paying the wages of the club,” said Cuddy.
“We don’t know if that is it. Jardine Norton could come back and it is not an ideal situation.
“Money is still owing and it will come back. That is a debt of Neath Rugby Limited and the debt will go with the club if it is sold.”
Cuddy says he is in negotiations with two groups about taking over the club in its current form, while there was a third option in the Neath Supporters Group.
During his 25-year involvement, he said he had invested millions and around £500,000 in the last three years.
“I am keen to hand it over to people who would run it as I would want it run and care about it as much I do,” said Cuddy.
“Time is of the essence and I want to exit the business as soon as possible. It could be weeks rather than months.
“We want to make sure it is adequately financed, which it has been for most of its history.
“We are in discussions with parties who have expressed interest, but it is a difficult time because of the events of the last few weeks.
“There is still an opportunity to sell the club. There are two parties interested in buying the club in its current form and they are both credible.”
Neath Supporters Club could still provide another option and they wanted the club, which trades as Neath Rugby Limited, to be wound up so they could take it over.
“We thought it was three parties, but one has withdrawn and that is the supporters group,” said Cuddy.
“They seem to have decided the best route for them is to start afresh and are waiting for the current club Neath Rugby Limited to fail.
“They appear to be confident they will be given the opportunity to run the club after that.
“But that will mean the club will be relegated a number of divisions, I would have thought, as opposed to the current Neath Rugby Limited which in all likelihood would go down one division if the club fulfils its fixtures this season.
“So I would prefer to sell the business in its current form because it would have a more certain future.”
Cuddy says his health stopped him from attending a supporters meeting earlier this month, but wants to meet a small section of the supporters group.
“I offered to meet them last week and they did not want to and I am sure that is for what they see as sensible reasons,” said Cuddy.
“It is difficult to engage in transferring a business or club if you don’t speak and meet the person that owns it.
“The offer is still there and I would be delighted to meet with them if they want to come and talk.”
Cuddy is one of the most recognisable figures of the Welsh regional rugby era and was a driving force behind the success of the star-studded Ospreys in the decade following its inception in 2003.
But he says his illness hit him hard.
“I spent six months in hospital and six months in bed at home,” said Cuddy.
“It took them about 15 months to work out what was wrong and that is because it is such a rare illness.
“I am a weaker person than I was and it affects a lot of my physicality.
“My mobility is not what it was and I am unstable on my feet.
“There have been comments recently when I go to the rugby and sit in the car, but that is not out of choice.
“I have always wanted to watch from the dugout and stand but I don’t feel stable enough.
“My life is not what it was, but hopefully I am getting stronger every day.”
‘Abuse comes with the territory’
Cuddy says his recovery has not been helped by abuse he has received.
“The last two years have been the most difficult of my life,” said Cuddy.
“It is not pleasant for anybody. It is upsetting for me and my family.
“You have to have a thick skin when involved with owning a rugby club because things can turn.
“I have been involved in rugby for some time and understand it comes with the territory to some extent.
“But they have been very personal attacks online and there have even been threats against me.
“Personally I can take it, but it is the effect on my family.
“It has been horrible for my family and friends who know what I have done for the rugby club over the years.
“But some former players have been very supportive and we have been getting messages from throughout the rugby world.”
Neath’s immediate concern will be to field sides against Bridgend on Saturday, 22 December 2 and Aberavon on Boxing Day.
The bulk of the squad along with the coaching staff have left and Neath have been relying on players borrowed from local clubs.
“It is difficult because the people who have been assisting have their own commitments and fixtures to fulfil,” said Cuddy.
“We are trying to get the fixtures on while trying to dispose of the club. Every effort is being made to try and fulfil every fixture.
“The club has a lot of friends and there have been a lot of offers to help with some threads on social media.
“It is likely we will be relegated, but we want to fulfil all fixtures.
“The former coaching staff chose to leave and you have to replace them. People have put their hands up to help, but it is hard establishing this in the middle of the season rather than at the start.”
Cuddy says there has been “little input” from the Welsh Rugby Union (WRU) and Ospreys while Neath Port Talbot council is an interested observer.
Neath Port Talbot council leader Rob Jones will meet WRU chairman Gareth Davies this week to discuss the club’s situation.
“We hope they will be supportive and understand we are doing everything we can to pass on the stewardship of the club,” said Cuddy.
“We are the oldest rugby club in Welsh rugby and hopefully that can buy you some time.
“Hopefully we can get back to where we were, but it will be a long road and depend on who takes over.
“Neath is loved in the community. Neath Rugby club and the Gnoll are synonymous with each other. One would have no significance without the other.
“The council knows how important that it is to Neath people and hopefully they will support Neath reincarnations in the future.”