Under former Celtic and Liverpool boss Rodgers, Leicester finished fifth in the Premier League last season, narrowly missing out on Champions League football.
The Foxes have continued to impress this term, winning six of their first eight matches, and Carragher believes Rodgers is the ‘best British coach out there’.
Rodgers came close to winning the Premier League while he was in charge of Liverpool and clinched two Scottish Premierships during his spell at Celtic.
Blues legend Frank Lampard and former Gunners midfielder Mikel Arteta are currently in charge of Chelsea and Arsenal respectively but Carragher reckons one of the London clubs could move for Rodgers when they next look for a manager.
Carragher, who played his final season at Liverpool under Rodgers, told MailSport: ‘Brendan Rodgers is the best British coach out there, no doubt.
‘Maybe you could argue that the Liverpool job came too early but I think he did a good job, coming closer than any manager to winning the League in 30 years.
‘He’s a fantastic coach, he’s done a great job at Leicester and I think his next job could be a Chelsea or an Arsenal.
‘Maybe it would be difficult to go back to Liverpool or to coach Manchester United, given the Liverpool connection, but I think he’ll be up for the top jobs in this country, maybe England too.
‘I’ve been really impressed with Leicester this season in that he’s actually changed tack in the big away games. He’s not thought, “I want to dominate the ball every place I go”. He’ll sit back and play counter-attack.
‘They’ve won every away game, the way he’s set up. And certainly what he did at Leeds [Leicester won 4-1], with all the talk of Marcelo Bielsa, was outstanding.’
Lampard’s Chelsea beat Newcastle United 2-0 on Saturday, while Arteta’s Arsenal played out a goalless draw at Leeds United on Sunday.