Former England captain Gerrard was Rodgers' skipper when the pair were at Liverpool.
The 36-year-old midfielder moved from Anfield to join the Los Angeles Galaxy but it was announced Tuesday he was leaving the Major League Soccer side.
Gerrard, who made it clear he wanted to continue playing, was immediately linked with a move to Parkhead.
"I never speculate or speak about players coming into the club. I never have done and don't intend to start now - that will always remain private.
"The only thing I would say with regards to Steven, he has been an incredible player. I had him for a number of years at Liverpool and he was absolutely brilliant with me."
The Northern Irishman added: "As a young manager going into such a great club, having the captain and real senior player there, to be able to work closely together with him and learn about the club and work together on the field was fantastic.
"He decided to go to America and I'm sure he has had a good spell there. Steven will sit down now and come back and decide what he wants to do.
"In terms of the speculation, I'm not sure where it's come from. But players coming into the club will always remain private."
Rodgers, who has already been reunited with Scott Sinclair, Dorus de Vries and Kolo Toure since joining Scottish champions Celtic, added: "I wouldn't tell you if there was anyway, but I suppose it's that time of the year now, we are getting towards the January window and Stevie's time is finished out there at Galaxy so I'm sure there is speculation over where he might go.
"And of course there's a natural look at Celtic because of the size of the club, and it's a club he admired, and also I have some of my former players here working with me. But we always keep our business behind closed doors."