The Northern Irishman has been out of work since being sacked by the Reds in October, replaced by Jurgen Klopp on Merseyside.
But speculation is rife that the Welsh outfit will turn to Rodgers (The Independent) if current manager Francesco Guidolin leaves the club when his contract expires this summer.
The 43-year-old came within a single point of guiding Liverpool to the Premier League title in 2014, and Rush says that Rodgers, who left the Swans to join the Merseyside outfit in 2012, would be returning to the Liberty Stadium a better manager as a result of increased experiece.
"I think Brendan is a good person and I like him," said Rush, as quoted by The South Wales Evening Post. "Things started to go wrong at Liverpool, but I think he'll learn from the experience.
"People always say don't go back, but I did it as a player and enjoyed it. I don't agree with that saying and there are many other people who have returned to clubs and done well. Eddie Howe has done it at Bournemouth.
"Brendan would come back to Swansea after a lot of new experiences and possibly be an even better manager than before. It depends what Brendan wants to do and what other offers are about, but it would be interesting if he did decide to do that."
Rodgers, who led Swansea to Premier League promotion in 2011 and helped lay the foundations which established them as a top-flight club, would certainly return to South Wales a better manager.
At 43, he has squeezed quite a lot into his relatively short managerial career thus far, be it challenging for major honours or dealing with difficult players, and City chairman Huw Jenkins may want him to continue his development at the Liberty Stadium.