Southgate is one of the leading contenders to take over. Peter Taylor and Phil Neville have also been linked with the job, as has ex- England chief Glenn Hoddle, says The Sun.
FA chairman David Bernstein - who retires next month to be replaced by Greg Dyke - refused to speak about Pearce's future.
He said: "That will be discussed after the tournament.
"He has been pretty successful for us but this tournament hasn't been a success.
"Overall, I think he's done well over a long period, particularly with all the factors. He's operated with quite a difficult hand."