Talks are still ongoing between the clubs, but the news comes on the same day that the Reds boss stated he was close to making a "real exciting" new signing.
The Merseyside club has been linked to a raft of new players, including Daniel Sturridge and recent Tottenham acquisition Gylfi Sigurdsson, but this would be Rodgers' first foray into the transfer market.
The 21-year-old Italy international scored nine goals in Serie A last season and has worked with Rodgers at the Chelsea academy and on loan at Swansea as the south Wales club were promoted to the Premier League.
When questioned over whether there would be any imminent news on signings, Rodgers told reporters at a press conference on Monday that there were one or two prospects being lined up, but that the club would not pay over the odds.
He explained: "There have been a whole raft of players being linked and only a few are real targets for us.
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"The reality is we have only made three or four enquiries about players and hopefully can close out one deal this week which will be a real exciting one for the club, one the supporters will absolutely love.
"There certainly won't be as many [signings] as is being labelled out there. We want to work with the great players we have here and make three or four tweaks in relation to the squad that can help the group."
Only last month the player was linked with a transfer to Anfield but quashed rumours by signing permanently at the Stadio Olimpico.
"I am at Roma now," Borini told Goal.com. "Roma have just bought me, either today or yesterday.
"It's not my business now to go to Liverpool or to stay. I don't know [if there is a possibility of doing so]. For now, I am a Roma player."