Enrique, 29, joined Liverpool from Newcastle four years ago, and whilst he played 78 games in his first two seasons at the club, he has featured in just 18 games since.
Now well down the pecking order under Brendan Rodgers having fallen behind Joe Gomez and Alberto Moreno, Enrique looks set to leave Anfield before the transfer window slams shut on Tuesday.
The Telegraph report that West Brom boss Tony Pulis wanted to sign Enrique earlier this summer, but couldn't get a deal done, meaning he looked elsewhere.
Now, with just over a day to go until the end of the summer transfer window, it's added that Enrique still has an offer to join West Brom this week, with Pulis keeping the opportunity to move to The Hawthorns open.
With Chris Brunt - normally a winger - filling in at left back until West Brom make a new signing, it's clear that Pulis needs to bring in a naturally left-sided defender, and Enrique certainly fits that bill as a proven top flight left back.
Enrique possesses Premier League experience, and with Liverpool seemingly trying to give him away in the final hours of the transfer window, West Brom could land themselves a bargain - if he can replicate his success from his days at Newcastle.