Malky Mackay confirmed his interest in bringing the foward back to the club.
It is understood Liverpool are reluctantly willing to let the Wales international leave Anfield for personal reasons.
Bellamy played for home-town club Cardiff on a season-long loan spell in 2010-11 after finding himself out of favour at Manchester City.
The 33-year-old joined Liverpool from City on a free transfer last summer and impressed during his second stint at Anfield.