The Welsh midfielder has looked excellent at times under Jurgen Klopp.
The 26-year-old appears to have improved quite a bit since the German replaced manager Brendan Rodgers in October, though first-team opportunities remain scarce for the Wales international.
But Thompson is a big fan of the Welshman - whose contract expires next summer - and urges the former Borussia Dortmund coach to offer him a new deal.
"One thing I want to see Klopp do quickly is get Joe Allen signed up on a new deal," he said to Sky Sports.
"He was brilliant against Bournemouth and whenever he's come on, against Dortmund and Everton, he's been terrific."
Thompson is right. Allen is certainly one of several players who look reborn since Klopp came in and a lot of Liverpool fans seem to have acknowledged his resurgence.
He has a good attitude and Klopp may just want to keep hold of him. The problem is that the 48-year-old may not be able to guarantee him regular football next term and Allen will know this, too.