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.
Swansea saw their £8 million bid rejected at the end of June, but they are wideley reported to be ready to match Stoke’s offer, giving the player a straight choice of both clubs.
Sevilla have also been linked with the player but it is understood the fee is beyond what they are willing to pay, while Manchester United have also been linked with an unlikely bid.