Allen left Swansea for the Reds in 2012, but after four seasons on Merseyside the midfielder is considering a move away from the club.
But according to BBC Sport, he may be offered the opportunity to return to the Liberty Stadium.
Swansea have already seen an £8 million bid rejected by the Reds earlier this summer, but following their takeover by an American consortium, they could renew their interest.
“Joe is one of our own players and if at all possible we would probably make that signing,” Swansea chairman Huw Jenkins said this week.
They would likely need to match Stoke’s offer, and for a squad player in the final year of his contract, this represents great business for Liverpool.
Allen spent 13 years on the books at Swansea, progressing through the club’s youth ranks and aiding their rise from League One challengers to established Premier League club.
Returning to south Wales could provide Allen with a more attractive prospect than turning out at the Britannia Stadium.
Swansea have already made five signings this summer, with the permanent acquisition of Dutch midfielder Leroy Fer their most high-profile business so far.
Allen could join Fer in a new-look midfield under Francesco Guidolin, restored to the key role that his ability demands.
“I feel good [about Joe]. I like the guy and I really think he is a wonderful football player,” he explained.
“I knew this before the European Championships, I saw it during the Euros and I know it now.
“I have said all I can say about this. It is all about respecting contracts, negotiating or not—whatever—and making a transfer or not.”
Allen is not due to join up with Klopp’s squad for their pre-season tour of the US, however, with a move away from Anfield likely before the end of the summer.