The former Porto boss is the favourite to become the next manager of the Toffees, as Frank Lampard is still in the running.
An announcement is expected in the next 24 to 48 hours.
The Spaniard was fired last week by the club.
The former Fenerbahce manager has held further discussions with the owner of the club after his initial meeting with him.
No contact has been made with Derby County with regards to Wayne Rooney, who has also been linked with a possible return to his club.
He was a candidate for the manager's job twice before, but missed out on the job.
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The Belgium FA don't want their head coach to leave ahead of the World Cup in November, so that's why the club didn't want him.
Duncan Ferguson could not guide them to a positive result on Saturday after a 1-0 home defeat against Villa.
The loss leaves the team 16th in the league.
The decision on a permanent successor to Benitez is expected to be made this week, with important games against Newcastle and Leeds on the horizon.
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He has no experience in the English top flight and spent three years in China with Shanghai SIPG, his longest spell at any one club.
If Duncan Ferguson was given the job until the end of the season, it would be a gamble, but if he was given the job only until the end of the season, it would be a risk, and the club would be in danger of being demoted from the top-flight for the
It would be the second time that Moshiri has gone against the wishes of the board, having appointed Benitez.
When Silva was dismissed two years ago, it was believed that he was the leading candidate before he removed himself from the application process.
The opportunity appears before him again, after he was under consideration byEverton in 2013).
The Turkish giants are currently fifth in the table and 14 points off the leaders, thanks to an unsuccessful stint by Pereira. Why is he in the frame again if he wasn't appointed in the first place?
His admirers will point to the fact that he has managed a number of top European clubs, including Porto, where he won back-to-back Portuguese top-flight titles.
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1860 Munich was demoted to the German third-tier in his six months as manager.
In his first season, he only won six of his 20 games, leading to recriminations over who was responsible.
It's not clear whether Pereira is the right person to take over at a time when the club are in the midst of an identity crisis and at the wrong end of the table.
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His previous experience in a battle for survival suggests he isn't the one currently needed. He will want to prove his best days in the dug-out are still to come. The international break in January gave the team some time to think about their next move.
The dilemma for Moshiri is that the manager he needs to get them out of trouble is probably not the one he needs to turn around the club's fortunes.
There is no director of football, head of recruitment or scouting at the club. It means that Moshiri will have final say.
He doesn't have to make snap decisions, but he has to make sure he doesn't get it wrong again.
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