Manager Paul Trollope told the club website: "We are delighted to welcome Mikkel back to the Memorial Stadium.
"He was in our thoughts throughout the summer, however it looked as though we would not be able to secure his services for a return."
Andersen was first choice between the sticks last season making 39 appearances and becoming a firm favourite with the fans.
Rovers were down to just one keeper, Mike Green, after terminating Luke Daniel's contract and sending him back to West Bromwich Albion due to a recurrence of his back problem.
Great news - shame it's only for a month though - hopefully we can keep him for the season.
Tangled Up In Blue
"In light of Luke's injury, and the current situation at Reading, we have been able to bring Mikkel here for the start of the season," added Trollope.
"Initially it is a month long loan, which will fall within the transfer window, so we will be in discussions with Reading throughout the month about the possibility of an extension."
Source: BBC Sport
Source: BBC Sport