The 26-year-old is popular with the Anfield faithful and starred for Wales at Euro 2016 as they made history by reaching the semi-finals.
But Allen has only 12 months left on his contract with Liverpool and has not been offered a new deal.
Reds boss Klopp is now understood to be ready to cash in on Allen this summer as he looks to reshape his squad.
Swansea, who sold the player to the Reds for £15m in 2015, are keen to re-sign their former star.
Southampton are also understood to be showing an interest, while Spanish giants Sevilla are monitoring his situaton having been impressed by his performances in France this summer.
Liverpool are believed to have slapped an £11m price tag on Allen’s head.