Vermaelen is his his second season with the La Liga side, but he has made only six La Liga starts this season.
He has made 14 appearances in all competitions, failing to win a regular starting berth, but he is at least fit to be considered to play, unlike last season.
The centre-back made only one appearance late in the campaign after being out injured ever since signing the previous summer.
A fresh start would likely do him good, and a return to the Premier League where he once starred for Arsenal is a sensible option.
Vermaelen previously captained the Gunners, establishing himself as a leader in defence and goal threat in attack, and could be useful cover for Liverpool, with Mamadou Sakho facing a lengthy ban.
Liverpool are already set to sign Joel Matip this summer, having agreed a pre-contract agreement, while Dejan Lovren finally hitting form makes a solution at the back less pressing.
Vermaelen would however be a quality squad player and is a Belgian international. The issue will be staying fit, and potentially getting through a medical.
Here is a look at how fans reacted to the news on Twitter...
Vermaelen, really? Like seriously... Come on now Liverpool. What do I know, I trust Klopp so... Yep. #LFC— Ed Iputo (@reddie_eddie) April 27, 2016
ThisIsAnfield claims Liverpool are willing to pay just under £8m for Thomas Vermaelen this summer #LFC would be a good squad player— StraightOuttaAnfield (@PureLFCTalk) April 27, 2016
Might as well forget about Sakho for a while then. Vermaelen not a bad cover, imo. #LFC— #8 (@KingBasu1) April 27, 2016
I think Vermaelen would be a perfect signing for #LFC— Spencer Tailford (@SpencerTailford) April 27, 2016