Talks are at an advanced stage with a consortium believed to be headed by former Wasps player Ken Moss, who has made millions from the IT industry.
It is understood the group also includes people with considerable sports commercial expertise, while a new stadium site has been identified close to the M40 just 10 minutes away from Wasps' existing Adams Park home in Wycombe.
Wasps, European champions in 2004 and 2007, faced the serious prospect of going into administration unless a buyer could be found.
But having secured their Premiership status, despite losing 14-10 at home to bottom club Newcastle, the immediate future looks a whole lot brighter than it might have done.
"The position with regards to the sale of the club is that terms are agreed, solicitors are instructed and we have entered into a period of exclusivity," Rigby said.
"We are now thrashing through the finer detail. As ever with a transaction of this type, there is always lots of detail to go through.
"I am pretty confident, particularly after securing Premiership rugby for next season, that the deal is firmly on track. I hope to see it concluded by the end of May.
"It is much easier having a discussion with a consortium when you can confirm you will be playing Premiership rugby next season. It is a major positive.
"At this stage, I am most interested in the sale of the club, rather than the business plan upon which they are buying the club. The business plan is there.
"I am just keen to ensure the club has a positive financial future."
Wasps thought they had found new owners last month, but that deal collapsed at a late stage, plunging the club into a period of huge uncertainty while they battled for Premiership survival.
"By the end of May, I believe we will be in a position to conclude transaction," Rigby added.
"I say that carefully, given the fact we were very close to concluding a deal previously. In that context, it is never done until the ink is dry on the contract.
"But I am pretty confident in the quality of the consortium and the quality of their interest.
"It is pretty certain we will be playing at Adams Park next season, but we need to resolve a number of detailed items, including where we are playing next season.
"As far as I am concerned, I see the glass half full. I am not even thinking about a scenario with the glass half empty.
"The starkness of our financial position was laid bare for everyone to see, but despite that I was very confident we would be able to piece together a deal that would secure the financial future of the club."
Rigby has also paid tribute to Wasps rugby director David Young, whose admirable perseverance amid an injury-ravaged campaign reaped its reward on survival Saturday.
"What we have been really pleased with has been Dai Young's contribution in keeping the squad pointing forward in a very positive way while this cloud of uncertainty over ownership has hung over the club," Rigby said.
Wasps' injury troubles followed them right through to the regular Premiership season's final weekend, and Young said: "I would be a liar if I said I wasn't nervous before the game.
"We had four or five plans if players didn't get through the warm-up. On Thursday, only 12 players were fit to train. It has been like that all season.
"It couldn't have gone any worse, either on or off the pitch this season. It's been very draining, both physically and mentally.
"It is disappointing not to have won (against Newcastle), but we've done enough to stay up and next season, with new signings, our young talent more experienced and surely a smaller injury list, it is looking more positive."
Wasps arrivals this summer will include England internationals Tom Palmer and James Haskell, who are returning to the club, Wales' most capped player Stephen Jones and South African Ashley Johnson.
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