Jurgen Klopp's centre-backs have suffered several injuries in recent months.
The Merseyside club's injury problems have been well documented this season, with The Guardian reporting earlier in the month that at one stage, all of their senior centre-backs - Dejan Lovren, Mamadou Sakho, Martin Skrtel and Kolo Toure - were out injured.
Those problems appeared to persuade Jurgen Klopp to make a surprise move for Queens Park Rangers defender Steven Caulker, who has moved to Anfield on loan until the end of the season (via BBC Sport).
Despite signing the former Southampton loanee, Thompson believes that the club still need to look at strengthening their back four in the summer, and he suggested that they should be looking towards the Britannia Stadium for the answer.
"When Liverpool evaluate at the end of the season and look at the sort of goals they conceded, I wouldn't be averse to see them looking towards Shawcross as a solution," he told Sky Sports.
"In fact I think he would be absolutely ideal and he's been one of the most underrated centre-backs over the last five seasons. You just have to look at Stoke's stats with and without him."
Following Saturday's dramatic 5-4 victory over Norwich, Klopp's side are currently 7th in the Premier League. However - perhaps unsurprisingly, given the result at Carrow Road - they have conceded more goals than any other side in the division's top 11, and currently have a negative goal difference.
The 28-year-old would arguably bring a presence that would make the Reds much more difficult to beat from crosses and set-pieces, which has been an area that they have looked particularly vulnerable this season.
His consistency would also bring a much-needed reassurance to the defence, and his unspectacular style may prove to be just what Liverpool need.