Brendan Rodgers opened negotiations with an offer of £10million plus Jonjo Shelvey on loan for the Team GB midfielder, but Swansea chairman Huw Jenkins turned the bid down.
The Welsh club are thought to value Allen at £15m, and a Liverpool bid to that figure would automatically trigger a release clause in Allen's contract.
Brendan Rodgers worked with the midfielder during his time as manager of Swansea, and the new Reds boss is very keen to make Allen his first Liverpool signing.
Reports suggests Allen is also keen to link up with his former manager, and Liverpool are likely to return with an improved offer.
Rodgers does have signed an agreement with Jenkins not to poach any of Swansea's stars after the Swans chairman gave Rodgers permission to join Liverpool, but the Welsh club reserve the right to waive that agreement in the best interests of the club.
Both clubs are currently on pre-season tours to America as they prepare for the new Premier League season, but Allen is with Stuart Pearce's Team GB squad for the upcoming Olympics.
The Swansea midfielder is renowned for his neat passing, and Rodgers is thought to view Allen as a crucial component to his new Liverpool side as he looks to implement his own footballing philosophy.
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