Swansea will step up their interest in the Icelandic international with a formal approach .
Rodgers, who worked with Gudjohnsen while on the coaching staff at Stamford Bridge, said: "I will need to make the call to Tony (Pulis) and see what develops.
"Eidur will want to play at the highest level he can, but I know his ability and what stage he is at in his career.
"He is pure quality and would be fantastic for this club. He has a great mentality and works hard in training.
"Only 18 months ago, he was with Barcelona so he would be great at this level.
"Eidur is still a Stoke player and would be a long shot for a club like us. There are lots of things that would need to happen before someone like him could come in.
"If there is any way we can do a deal for a player like that then I would always look to see what we can create."