Earlier this month Goal.com revealed Brendan Rodgers was ready to make signing the 22-year-old a priority after securing the services of Fabio Borini, who is expected to make his debut against his old club Roma in a showpiece friendly at Boston's Fenway Park on Wednesday.
Allen proved integral part of Swansea's impressive debut Premier League season, making 36 appearances and his impressive performances helped him secure a place in Stuart Pearce's Team GB squad, which takes on Senegal at Old Trafford on Thursday.
The Welsh club ignored the contractual clause agreed when Rodgers moved to Merseyside, which was designed to ensure that no Swans players could be signed by Liverpool for a minimum of 12 months.
Swansea had hoped to command a fee of 15 million pounds, which would have activated Allen's release clause, but has agreed to a slightly lower valuation following last week's arrival of Spanish midfielder Michu from Rayo Vallecano in a deal worth 2 million pounds.
Liverpool hopes to finalize the transfer before the end of this week, ahead of its friendly with Tottenham in Baltimore. Gylfi Sigurdsson, who earlier this month snubbed a reunion with Rodgers, is expected to feature for Andre Villas-Boas' side.
Meanwhile, Goal.com understands that Liverpool's attempts to sign Clint Dempsey remain at an impasse, with Fulham reluctant to part company with last season's top goalscorer.
Rodgers last week confirmed Liverpool's interest in the American, who has a year remaining on his contract with the west Londoners, but insisted that only initial enquiries had been made.
He also intends to trim his squad ahead of the new season with goalkeeper Alexander Doni expected to secure a return to his native Brazil after being deemed surplus to requirements following just one season and a solitary senior appearance.
The 32-year-old, who arrived from Roma on a free transfer last summer, was omitted from Liverpool's squad for its tour of the United States due to personal reasons, but with Fenway Sports Group looking to trim their excessive wage bill, his future lies away from Anfield