Town boss Nathan Jones could be forced to act following the expected departure of Croatian international Simon Sluga, who will be out of contract in the summer, with Talksport report Alex Crook stating the 25-year-old will be on his way to Kenilworth Road.
Palmer began his career at the Hawthorns, turning pro in May 2015, and has played one League Cup game for the Baggies, that coming in a 6-0 defeat to Arsenal this season.
He went to Kidderminster on loan in July 2015 until January, making 11 starts, before brief stints at both Oldham Athletic and Notts County, playing once for each side.
In July 2019, he went to Plymouth Argyle, with 41 outings for the Pilgrims, while last season, Palmer was with League One Lincoln, racking up an impressive 58 appearances.
The keeper, who has also represented England at U16 level on four occasions, is currently second choice under Valerien Ismael, promoted to the bench after England international Sam Johnstone has missed out recently due to an internal issue.