New Everton owners, the Friedkin Group, are said to be considering manager Sean Dyche's position.
According to talkSPORT, the Friedkins are weighing up bringing in their own man ahead of completing their purchase of Everton from Farhad Moshiri.
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Three names are being considered - former Chelsea boss Graham Potter, ex-England coach Gareth Southgate and David Moyes, who is available after leaving West Ham over the summer.
Like former Toffees manager Moyes, Potter and Southgate are also immediately available as free agents.
With just one point from the first five games, Everton are in 19th place on the Premier League table.