As Goal.com exclusively revealed two weeks ago, Rodgers turned his attentions to Allen after completing the signing of Fabio Borini, a player who had also worked under him during his time with the Swans.
And, despite reports the 39-year-old had agreed not to poach players from his former club, Rodgers insists he would be open to signing the Wales international.
He told reporters: "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
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"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 on-going negotiations on a few targets."
The former Reading boss also revealed his desire to bring in further new faces before the end of the transfer window.
"We hope we're close to some deals,” he added.
“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."