Paul Ince left the Dons to take over at Ewood Park last month and is keen to be reunited with the 27-year-old midfielder.
A statement on MK Dons' official website read: "In order to prevent the intense speculation of recent weeks jeopardising the start of our season, we feel it both in the club's and the Keith's interests to settle this unhelpful situation.
"Keith is a player we value extremely highly and remain committed to retaining, but we maintain our policy of not standing in the way of an individual's professional development.
"Despite widespread reports linking him with many clubs, we have received no formal offers for Keith. This action has been mutually agreed by the manager and the chairman."
Andrews has scored 21 times in 85 appearances during his two seasons with the Dons.