Liverpool have accepted a £13m offer from Stoke for the Wales midfielder.
But Americans Jason Levien and Steve Kaplan want to bring the former Swansea star back to the Liberty Stadium after their £100m takeover.
Levien said: “He [Allen] is certainly someone we have our eye on and we are discussing making a bid.
“We spent time thinking how additions to the club will fit in with the structure and he is someone with a great history here.
“He is still a young player. He is someone who multiple clubs are interested in, us included.”