Craig Bellamy has pushed for a move away from Liverpool.
The Guardian says Bellamy is closer to an Anfield exit and could complete a return to Cardiff City within days.
Rodgers had wanted to keep the 34-year-old at Liverpool but the club has reluctantly granted the striker's request to be closer to his family in south Wales.
Queens Park Rangers are also keen on the Wales international, who is now available to complete a transfer following Team GB's elimination from the Olympic Games, but his preference is to rejoin his hometown club where he spent 2010-11 on loan.