Allen could move to Liverpool soon
Liverpool manager Brendan Rodgers has long been a keen admirer of the 22-year-old, and the Welsh club had already rejected an offer worth £12.5million.
But former Swans boss Rodgers was always expected to return with an improved bid, and Press Association Sport understands that, while a formal offer has yet to be made, Liverpool have verbally confirmed their intention to stump up the money required.
Swansea manager Michael Laudrup is to hold talks with the player over the interest from the Merseyside giants, but it is understood that Allen is keen to make the switch to the Reds.
Swansea and Liverpool have a written agreement that prevents Rodgers from trying to sign any of his former players over the next 12 months.
But that clause can be waived if another club shows interest in a Swansea player, or if the Swans deem a deal to be in their best interests.
Allen would be Rodgers' second signing since taking over at Anfield following the arrival of Fabio Borini, himself a former Swansea player, from Roma.
Laudrup had made it clear that Allen, who returned to training from Olympic duty on Wednesday, would ultimately decide his own future, and is keen for a resolution to be found quickly to allow him time to bring in reinforcements before the end of the transfer window.
Bolton midfielder Mark Davies is understood to be one of the players interesting the former Barcelona and Real Madrid midfielder, who is also close to bringing in striker Itay Shechter on a season-long loan from Kaiserslautern.