The 22-year-old had talks with Swans chairman Huw Jenkins on Wednesday and will meet with manager Michael Laudrup on Thursday.
But Liverpool, under the management of ex-Swansea boss Brendan Rodgers, are confident they can proceed quickly to complete the deal.
The two clubs opened negotiations over a deal for Allen last month, but the Reds had their opening bid rejected.
Rodgers admitted his interest in Allen last month when the Irishman said: "He is a very talented player, which everyone knows."
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 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.
Allen returned to training at Swansea on Wednesday following his involvement in the Olympic football tournament with Team GB.