The Sun says the Reds may have to fight bitter rivals Manchester United for Stokeâ€™s Bosnia international keeper.
Rodgers sees Begovic as the long-term successor to Pepe Reina, who had been in far from convincing form before getting injured, although he returned to the team for last weekendâ€™s 3-0 win over Wigan.
Brendan Begovic is on Liverpoolâ€™s shortlist of possible replacements, with Rodgers also known to be a fan of Birminghamâ€™s young England star Jack Butland.
But Begovic is also on the radar of United boss Alex Ferguson.
The United boss is still not sure about David De Gea and Anders Lindegaard.