Allen has grown into a prominent role in the Reds’ midfield this season, and has hit a consistent level of top form under Jurgen Klopp, but doubts remain over his long-term future.
The 26-year-old’s contract is set to expire at the end of 2016/17, with no talks forthcoming at this stage.
Swansea made a loan bid for Allen back in January, with Klopp vetoing the move due to injuries throughout his first-team squad, but this has not deterred the south Wales side.
According to the Mail, the Swans will step up their interest by launching a bid for Allen this summer.
Swansea are said to be “huge admirers” of Allen, who spent 13 years with the club before making a £15 million move to Merseyside in 2012, and he would certainly thrive in their midfield.
Current manager Francesco Guidolin has utilised Jack Cork, Leroy Fer and veteran Leon Britton in regular roles since his arrival at the club, but this has offered little in terms of stability.
Allen’s technical quality and tactical intelligence would be a major boost to the Liberty Stadium side.
Swansea are said to have baulked at Liverpool‘s asking price at this stage, however, with the Reds looking for “upwards of £10 million.”
Klopp may look to keep Allen at Anfield beyond the summer, either way, with the 48-year-old discussing his contract situation earlier this month.
“He deserves a future here, of course. He is in brilliant shape,” he said.
“We will see what happens but for me it is not the time to talk about his contract now.”
The Reds can still secure a place in the Europa League final with a win over Villarreal in next weeks’ semi-final second leg, after losing 1-0 at Estadio El Madrigal on Thursday night.
This will likely see Allen take up a starting role, and his performances before the end of the season could aid his cause in sealing a contract extension.
With a year left on his current deal, there is no immediate pressure for Liverpool to offer a new contract, but it is likely Allen’s situation will be resolved before the start of 2016/17.