Rodgers narrowly missed out on the Premier League title during his three years in charge of the Reds before he was sacked in October 2015, clearing the way for Jurgen Klopp to take over.
While Liverpool have improved drastically since then his departure, Rodgers went on to enjoy a successfull spell at Celtic and has had a big impact at Leicester since taking over in March this year.
The Foxes are third in the Premier League after 10 games, eight points behind leaders Liverpool and seven ahead of Manchester United, whose struggles continue under Ole Gunnar Solskjaer.
Despite his good work in British football, former Liverpool centre-back Carragher believes ex-Swansea boss Rodgers has no chance of taking over at Old Trafford in his career, but London could be the right place for him.
"Liverpool is out of the question and Manchester United is too because he was Liverpool manager," Carragher is quoted as saying by The Express.
"You look at Arsenal and Chelsea in a few years' time, whether Frank stays there or not.
“Rodgers has a link to Chelsea, being in the youth set-up and the reserves.
“They're the two jobs you'd look at, maybe international wise as well."
England have progressed during Gareth Southgate’s three years at the helm and the 49-year-old’s contract runs until 2022, but Carragher already sees Rodgers as a potential replacement.
He added: "Who knows how long Gareth Southgate will last for England, whether he goes after the next tournament or the next World Cup.
“I think Brendan Rodgers would come into the reckoning because there's no doubt, when he was flying at Liverpool, his name was continually mentioned for the England job.
“They're the types of jobs he'll be looking at but he knows that getting back into the Premier League after what happened with Liverpool, if he could get Leicester into the top four, it gives himself a great chance. It restores his credibility that maybe went a touch in that last season at Liverpool."