Andrews was Ince's captain as the Dons won promotion from League Two as champions and secured the Johnstone's Paint Trophy last season.
However, his future at stadium:mk had been in doubt after he informed new manager Roberto Di Matteo of his desire to play at a higher level.
Andrews has been linked with a move Rovers since Ince replaced Mark Hughes in the managerial hot-seat earlier in the summer.
The Irish midfielder, who began his career with Wolves before moving to Milton Keynes in 2006, was also wanted by several Championship clubs.