Fernando Torres will join Milan permanently on January 5 ahead of his expected return to Atletico Madrid.
Liverpool boss Brendan Rodgers has insisted Mario Balotelli will not be going out on loan in the January transfer window.
Olofinjana has found playing time hard to come by since returning from a serious knee injury that put him on the sidelines for nearly a year.
He has only made 39 appearances for the Tigers since signing for a fee of £3m in the summer of 2009 but did have a successful loan spell with Cardiff, where he scored six goals in 39 games.
The Nigerian midfielder is in line to make his Owls debut against local rivals Barnsley at Hillsborough on Saturday.