It emerged on Tuesday that Everton had reached an agreement with the north east club for goalkeeper Pickford.
The 23-year-old is currently on international duty with England U21s ahead of the European U21 Championships in Poland.
It was initially claimed that Everton were willing to wait until the end of the tournament before concluding a deal.
However, it now appears Everton are keen to wrap up the deal before England’s opening game against Sweden on Friday.
It is believed most of the medical took place on Wednesday, with the remainder of the tests due to take place on Thursday.
Personal terms are already thought to have been agreed, meaning a deal could be confirmed by Friday.
Should Pickford complete the move to Goodison Park he would become Everton’s record signing and the third most expensive goalkeeper of all time, behind Gianluigi Buffon and recent Manchester City purchase Ederson.
He would also become the club’s first signing of what we all hope will be a busy summer, as Koeman and Steve Walsh continue to re-shape the squad.