The former Middlesbrough boss confirmed after England’s World Cup exit to France that he is considering his long-term future with the Three Lions.
Southgate’s contract runs until 2024, meaning he would see England into the European Championships.
However, Southgate says the last 18 months have been difficult and he was stung by criticism after the side’s 4-0 defeat to Hungary in the Nations League.
England want Southgate to continue in the role but he has reserved the right to leave on his own terms after a successful five-year period at the helm.
The FA are therefore drawing up a shortlist of possible replacements and the Daily Star say Rodgers heads the list.
The Northern Irishman has a fine track record working with young players and his attacking philosophy would be a welcome change from the football England have played under Southgate.
Rodgers came close to leaving Leicester in the summer when the club refused to back him in the transfer market and he would be interested in the England job.
The likes of Steven Gerrard and Frank Lampard have been dismissed as candidates, while Graham Potter is out of reach.
Eddie Howe has long been thought of as an option but he is happy at Newcastle and wants to remain in the north-east.
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