Dempsey has been heavily linked with a move to Anfield throughout the summer after jumping to the top of the Northern Irishman’s shopping list after missing out on Gylfi Sigurdsson to Tottenham Hotspur.
The 29-year-old American international struck 22 goals for the Cottagers last season and manager Martin Jol is hoping he will sign a new deal as he attempts to fend off interest from the Reds.
It will be difficult task after Rodgers admitted to the clubs official website that he wants to sign Dempsey and the club have already set the wheels in motion.
“Clint is a player we’ve enquired about, it is as simple as that,” he told Liverpoolfc.com.
“Ian Ayre, our managing director, has spoken with the club to see what the position is. That is where we’re at.
“He’s a very talented player but we don’t like to talk about other clubs’ players.”
Rodgers also shed some light on Andy Carroll’s future at Liverpool with former club Newcastle United having a loan bid for the England striker rejected at the weekend just 18 months after packing him off to Merseyside for £35 million.
“There has been a lot written and spoken about him but first and foremost Andy is a Liverpool player,” Rodgers said.
“To consider a loan period for someone the club spent £35million on isn’t something we’re looking to do at this moment in time.
“Andy will be the same as every other player – if there’s ever an offer that comes in we’d look at it as a club and see if it’s going to be worthwhile for the club and the team as a whole.
“The club invested £35million in him. People talk about whether he can fit into my style or not, but if you’re a club and you spend £35million on a player you’d like to think he can fit into whatever style the team plays.
“He’s a good player. He’ll join the rest of the group next week and we’ll take it from there.”