Rodgers, who made the switch from the Liberty Stadium to Anfield to replace Kenny Dalglish this summer, confirmed his interest in the Welsh international after admitting he is also interested in Fulham's Clint Dempsey.
Allen signed a four-year deal with Swansea last year but Rodgers is keen to bring him to the north west. Rodgers told SkySports.com: "I have a great relationship with Swansea but if there is a market for any of the players, I would like to be in that market."
He added: "They respect that. He is a very talented player, which everyone knows but there is nothing more at the moment.
"When players come to a club such as Liverpool, it's never straightforward.
"There are always other barriers and difficulties. At this moment, there are ongoing negotiations on a few targets."
Rodgers has already signed former Swansea striker Fabio Borini in a £10 million deal with Roma and is keen to wrap up some more early business.
Rodgers said: "We hope we're close to some deals. I need to evaluate the players we have and assess the squad and bring some in to bring freshness to the group.
"As a manager you want them in sooner rather than later but the main thing is to get them in before the beginning of the season. If we do that, our job will be done."