Marcus Williams, Sam Togwell, Kenny Milne, Cliff Byrne, Matty Sparrow, Grant McCann, Paul Hayes, Jonathan Forte, Sam Slocombe, Matt Godden, Niall Canavan and Rory Coleman have all been asked to stay.
United would expect youngsters Slocombe, Gooden, Canavan and Coleman to sign as a formality, while they have already announced captain Byrne's acceptance of a two-year deal.
Milne, whose time at Glanford Park has been restricted by injury, has also gone on record indicating his desire to remain at Glanford Park.
The big question marks are over Williams and Hayes, with Williams previously making it clear he would like a move elsewhere and Hayes in demand after completing a clean sweep of the main player of the season awards.
Defender Picton and winger Winn came through the ranks at Glanford Park, but were only rarely involved in first-team duties.
May, a striker, was purchased from Millwall in January 2008, scoring five goals in 51 appearances. He fell out of the first-team picture this season, playing in just one game.