The former Manchester United star could be available for a knockdown price, as he continues his recovery from a second knee ligament injury.
Rossi will not be available for selection until March, but that will not deter the Anfield boss from making a bid, with Luis Suarez currently Liverpool's only recognised option up front.
Rodgers might be prepared to take a punt on the Italian international - having already made clear his priority to strengthen in attack in the New Year - in the hope Rossi can help fire the Reds up the Premier League table in the crucial final third of the season.
"We're looking to do business in the striking department in January," the Northern Irishman told talkSPORT over the weekend. "The reality is that there was always going to be a bit of movement this season with the club trying to regain some of their money. This season was always going to be difficult.
"We'll look to do something in the January transfer window, though. There's not going to be wholesale changes in terms of players coming in, that's the reality of where the club is at.
"We're looking to grow over the next six months, we'll get some reinforcements in January and that will support the group who have been magnificent for me so far. Hopefully in the summer we can build again."
Liverpool are believed to have already spoken to Villarreal, who may be forced to sell Rossi to ease their growing cash problems.
The 25-year-old has not played a league game in 13 months, but is still attracting interest from a number of clubs with reports in Italy suggesting Inter Milan have already agreed a deal with the forward.