Logan, who came through the Manchester City youth ranks with former Dons loanee Javan Vidal, will be out of contract with Brentford in summer.
The 26-year-old joined the Bees from City in 2011 after loan spells at Grimsby, Scunthorpe, Stockport and Tranmere.
Logan was a team-mate of Dons striker Niall McGinn two seasons ago when McGinn was on loan from Celtic.