The 24-year-old is being released by Forest boss Paul Hart as part of the club's squad downsizing and is on the hunt for a new home.
Posh manager Barry Fry is extremely keen to snap up the former Leeds United player, who can operate at full back, in midfield or in attack.
Fry, though, admits he may face stiff competition to land Gray as there is a limit to the funds he has available, if the battle to gain his signature turns into an auction.
"He would be a magnificent capture for us and at a time when almost every club is cutting staff, anyone who can play in two or three positions is a massive asset," Fry told the club's official website.
"I have spoken to Andy with his agent but am limited to what I can go to in terms of wages.
"There is no way I can become involved in an auction and obviously a number of clubs are also interested in the player, some of them from the First Division.
"My offer is on the table and in the meantime I am also starting to look elsewhere."
Meanwhile, Forest chief Hart was quietly impressed by the contributions of Lee Miller and Mark Kerr, who are currently on trial with the club.
The Falkirk pair played in Forest's 2-0 friendly defeat against Le Mans with Hart pleased by their performances.
Hart said: "I thought they acquitted themselves very well. The lad up front, Miller, played the whole game and Kerr got on for the second half."
