Everton are making a deadline-day bid to sign a goalkeeper, with Manchester United's Sergio Romero and Tottenham's Paulo Gazzaniga under consideration.
Manager Carlo Ancelotti said a new goalkeeper was not a priority last week but Pickford's latest error, which cost a goal in the 4-2 win against Brighton on Saturday, has prompted a rethink.
The 26-year-old has made a number of mistakes since the restart in June.
Romero, 33, is third choice at Manchester United behind David de Gea and Dean Henderson and would welcome a move to Everton, while 28-year-old Gazzaniga has been pushed down the pecking order by Joe Hart's arrival at Spurs.