The FA have drawn up a contingency plan in the event Southgate leaves at the end of Euro 2024 as he has strongly hinted.
Carsley is held in high regard after leading the under-21s to Euro glory last summer and he turned down the Republic of Ireland job earlier this year.
Other candidates would include the out-of-work Graham Potter and Newcastle boss Eddie Howe with the FA’s preference to appoint homegrown talent.
The FA insist they have not spoken to any potential successors and their desire is for Southgate to lead England to the next World Cup in 2026.
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