The Wales midfielder, currently on leave after his exploits in getting Wales to the semi-finals of Euro 2016, is in the final year of his contract.
Allen is looking for more playing time, having been restricted to mainly Europa League appearances last season.
Stoke's bid is roughly in line with Liverpool's valuation of a player with less than 12 months remaining on his contract - and would recoup most of the £15million they paid Swansea four years ago - so the decision is now likely to rest with the player himself, who is not due back to pre-season training until July 26.
Liverpool manager Jurgen Klopp admits he is an admirer of Allen but he refused to be drawn on the future of the midfielder.
MagistaX Proximo orange "There is nothing about this and all the transfer rumours," said Klopp after a 2-0 friendly win over Huddersfield.
"We cannot comment because we couldn't do anything else all day so we shouldn't start with this.
"I feel good (about Joe). I like the guy and I really think he is a wonderful football player.
"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.
elastico superfly orange "But it is nothing to discuss in public but I've not seen him since the middle of May in Basle (when Liverpool lost the Europa League final to Sevilla)."
Defender Jon Flanagan was left out of the squad to consider whether he wants to join Burnley on a season-long loan while Lucas Leiva, who is a target for Galatasaray, limped off in the second half against the Terriers with a hamstring problem.
neyjar jordan cleat He will have a scan on Thursday but it is likely to mean he will miss the tour to the United States, which the squad fly out to in the early hours.
Mercurial Superfly V orange Sadio Mane, a £30million signing from Southampton, was lively, having a goal harshly ruled out for a foul on Joel Coleman after a misplaced backpass and then inducing the goalkeeper to foul him for a penalty which Philippe Coutinho subsequently missed.
But it was midfielder Marko Grujic, only just turned 20, who impressed the most in central midfield and scored for the second successive pre-season game, rifling home Mane's cross after the forward had been sent free down the right by Dejan Lovren.
The second half was somewhat surreal with play briefly halted by a drone which appeared over the pitch before, even more bizarrely, Klopp sent on reserve goalkeeper Shamal George to play up front when Lucas went off injured after all outfield substitutes had been used and he almost set up Lazar Markovic with his first touch.
When Liverpool won an 89th-minute penalty the away fans chanted for George to take it but Alberto Moreno assumed responsibility and scored.
"The game was difficult. The first half we should have scored more," added Klopp.
"We missed a penalty and had a few big chances and played some wonderful combinations so it was really good.
ACE 16+ PureControl "But I was not too happy with the concentration level in the first 20 minutes.
"I said to the boys it was not bad but if we trust a little bit too much in our skills than a team performance it is not the best decision to make."
On the decision not to allow George to take the penalty, Klopp said: "Even though it doesn't look like it all the time these games are quite serious - we don't want to make joke games.
"It is nice when the crowd calls it in but at 1-0 in no game in the world should you make a 'funny' decision.
"If it is 4-0 or 5-0 you can think about it but even then it is a lack of respect for the opponent."
