Allen has fallen out of favour at Anfield, and new manager Jurgen Klopp has not started the Welsh international in either of the Premier League matches he has taken charge of so far.
There is a feeling that Allen is too slow to play in a team which is managed by Klopp, who demands constant movement and pressing.
Allen has played just three times in the Premier League for the Reds this term, and Leicester and Southampton reportedly feel that he could be available in January.
Leicester and Southampton have both made big strides in recent years, and their start to the season has seen them claim spots in the top-half of the Premier League table.
The pair are keen not to rest on their laurels though, and they feel that Allen would increase the quality of their midfield options.
Leicester and Southampton are not the only interested clubs in signing the former Swansea City midfielder, though, with Stoke City also said to be chasing his signature.