The Aviva Premiership club expect the takeover to be completed by mid-July.
Moss, an IT entrepreneur who played for Wasps between 1985 and 1990, stepped in to save the club after takeover talks with a different party collapsed in April.
The takeover would end Steve Hayes' unsuccessful four-year ownership of the club.
A club statement said: "London Wasps are delighted to confirm that contracts have been exchanged on the sale of the club to a new consortium headed up by former player Ken Moss, with full completion expected to take place in mid-July.
"Full details of the new ownership and future plans will be outlined once the deal is fully completed.
"The club's priority is now on ensuring the deal is complete within this timeframe as well as finalising arrangements for the club's future structure and operations."
Hayes put Wasps up for sale last July when Wycombe district council withdrew its support for the construction of a new stadium.
Moss will have to absorb annual losses in excess of £2million and repay a £1.5million secured debt.