Allen has emerged as a top target for former club Swansea City this summer, it is claimed, while Watford have also been linked with a move for the Welshman, along with Spanish giants Sevilla.
Allen has only a year left on his Anfield contract and Jurgen Klopp, the Reds boss, confirmed earlier this month that discussions over a new one are not forthcoming.
"We will see what happens but for me it is not the time to talk," Klopp is quoted as saying by the Mail. "He is in brilliant shape at the moment and deserves a chance. But contract talks at the moment? No."
Allen has been used frequently in the absence of Emre Can, and has repaid his manager with a string of impressive performances.
But the arrival of Serbian teenager Marko Grujic at Anfield could put the 26-year-old's place at risk.
Liverpool paid a reported £15 million to sign Allen from Swansea in 2012, a move which reunited him with former Swans boss Brendan Rodgers.
According to the Mail, Rodgers is preferred choice to replace interim head coach Francesco Guidolin at the Liberty Stadium this summer, and another reunion is in the offing.