The Welsh winger impressed for Team GB as they reached the Olympic Games quarter-finals but could be set for an Anfield exit, with the Ulsterman revealing Bellamy is looking to settle down as he approaches the end of his career.
"We have been in talks with him and Cardiff," Rodgers told reporters. "I had a really good meeting with Craig before I moved up here.
"He is a big Liverpool supporter but I also understand his thinking. He is 33 and has travelled around the country all his life, so there are compassionate reasons.
"We will look at it but we will wish him all the best and move forward."
Despite Rodgers' claims, Bellamy refused to be drawn on the speculation, insisting that he had no pressing thoughts on the future.
"Look, you know me better than to expect me to talk publicly about that. I've no immediate plans at the moment. What will be, will be." Bellamy told the South Wales Echo.
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"I don't think too far into the future. At this stage of my career I just have to take things day by day."
Malky Mackay is said to want to complete the signing as quickly as possible with his side getting their Championship campaign underway against Huddersfield in just over a week's time and the signing of the Wales international would give his team an added boost.
The 33-year old spent the 2010-11 campaign on-loan with the Championship club where they narrowly missed out on promotion to the Premier League following a semi-final play-off defeat but could now spearhead a new charge to the top flight.