Bellamy has been linked with a return to Cardiff City.
But the Liverpool Echo says Rodgers held talks with the 32-year-old earlier this week and believes he still has a great deal to offer the Reds.
"I had a really positive chat with Craig on Monday," Rodgers said. "It was good to meet him and get his side of the Liverpool story. He's a big supporter of this club - he loves the club.
"He's a good guy. He has a real passion for Liverpool Football Club and that's the type of people I want here."