The Argentine parted ways with Chelsea this week after guiding the team to a sixth place finish.
The Sun says Pochettino would be open to succeeding Gareth Southgate as England coach should the latter choose to step down after the Euros.
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FA technical advisor John McDermott worked with Pochettino at Tottenham.
Pochettino has said of the England job in the past: My relationship with England always been good.
"You never know what happens. I am open to everything."