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Ainsworth was given the offer of a one-year deal by Blues boss Gary Waddock on Wednesday morning and is thinking it over.
The 38-year-old is one of five players to be offered a new contract for the 2012/13 campaign.
Anthony Stewart is considering a new two-year deal while Matt Bloomfield and youngsters Jesse Kewley-Graham and Josh Scowen are mulling over one-year contract offers.
Striker Ben Strevens is the big-name departure from Adams Park. Anthony McNamee, John Halls, James Tunnicliffe, Steve Arnold and Jason Mooney have also been released.
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