The 33-year-old will become a free agent at the end of the month after Burnley announced they were not renewing his contract.
After a turbulent spell in his career, the shot-stopper – who has 75 England caps – is searching for a new club and Ricketts has claimed Salop enquired about his services.
Despite the approach, Ricketts insists Hart intends to continue his playing career at the ‘highest possible level’ but hinted that he may consider coming back in a non-playing capacity in the future.
“In terms of interest, from our part yes,” Ricketts said. “Joe would have an interest, but not so much as in coming back to play.
“He’s still at a level where he can play at a much higher level, not just in this country but abroad as well.
“It’s something that I put to him and his people, I said ‘do you fancy coming back’, but he’s ready to continue his career at the highest possible level.
“He has interest from top level teams so it’s not going to be one for us, sentiment doesn’t rule the head on this one.”
Shrewsbury-born Hart began his career at his hometown club before moving to Manchester City in 2006.
After several loan moves, he established himself as the club’s number one and went on to win several trophies, including two Premier League’s and four Golden Glove gongs.
His career stalled in 2016 when manager Pep Guardiola arrived in Manchester – as he was sent out on loan to Italian club Torino and West Ham – failing to secure permanent deals at either club.
He moved to Burnley in 2018 after injuries to Tom Heaton and Nick Pope, starting the 2018/19 season as first choice goalkeeper.
He was dropped for a recovered Heaton after a 5-1 loss to Everton on Boxing Day and failed to reclaim his spot.
Pope has since staked his claim as first choice and in the club’s first game back amid the coronavirus pandemic on Monday, Hart failed to make the squad against former club City, with Bailey Peacock-Farrell taking a place on the substitutes’ bench.
In the goalkeeper department at Shrewsbury, veteran Joe Murphy has been released, while Max O’Leary has finished his loan from Bristol City.
Former Wolves goalkeeper Harry Burgoyne and youngster Cameron Gregory have been offered new deals.