Winger Walcott scored a hat-trick in Arsenal 's sensational Capital One Cup win over Reading on Tuesday and Rodgers can offer him the striker role he craves.
The Reds boss is monitoring Walcott's continued contract impasse with the Gunners, and will move for the England international in January if he does not commit to a future at the Emirates, the Daily Mirror reports.
But Gunners boss Arsene Wenger claims he has not even considered selling Walcott in January as he believes the player will agree to stay, but admitted that Arsenal now urgently wants the England star to sign a new deal, the report said.
"There is urgency in sorting out the new deal - how much I don't know. But there is urgency. We want to sort it out before Christmas, one way or the other. I wouldn't like to sell him in January if he doesn't sign a new deal. I've not even thought about that, because at the moment I think we will still manage to make a deal with him," Wenger said.
Walcott's Arsenal deal expires next June and Rodgers hopes to buy him for a knock-down price when the winter transfer window opens, the report added. (ANI )