Rodgers has already brought in Fabio Borini from AS Roma and reports on Wednesday suggested they had secured a deal for Fulham's Clint Dempsey.
Allen was a key member of the Swans' squad during Rodgers' spell at the Liberty Stadium and his creative presence is thought to be high on the Reds wish list as they look to build on last season's eighth-placed finish.
The Northern Irishman has confessed his admiration for Allen, but stated that no deal is in the offing.
"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," Rodgers said on Allen during Liverpool’s pre-season tour of America.
"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.
"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."
