Taylor was told last week that he does not feature in manager Gary Rowett's future plans and that he is free to leave the club.
The 29-year-old, who joined Albion on a free transfer in the summer of 2011, has not made a first-team appearance since the FA Cup third-round defeat at Leicester on January 5, when he came on as a substitute after 62 minutes. His last start came in the league loss at Barnet on December 21.
Taylor will be available for Grimsby should they be involved in the play-offs. The Mariners are currently fifth in the Conference and occupy the final play-off spot.