Wigan boss Roberto Martinez is one of the names definitely in the running and remains the favourite but the identity of the other two is unclear at this stage.
Everton officials have recently spoken to Red Bull Salzburg sporting director Ralph Rangnick, who had success at German sides Hoffenheim and Schalke, but it is not known whether he has made the final trio of candidates to replace Manchester United-bound David Moyes.
Porto's Vitor Pereira has also been approached by the Toffees while the likes of Cardiff's Malky McKay and Celtic's Neil Lennon were also linked with the post.
Everton began with a list of about 30 names and have interviewed internal candidates, turning their attentions to out-of-work managers or those whose contracts are expiring - like Pereira - before lastly sounding out those currently working at clubs.
An approach for Martinez was made last Friday - and granted by Dave Whelan - and the Spaniard emerged as the favourite after telling the Wigan chairman on Tuesday he wanted to leave the FA Cup winners after four seasons as he did not feel he was able to fulfil the task of trying to get the Latics back into the Premier League after relegation.
Everton chairman Bill Kenwright met with Martinez in London on Wednesday and is expected to complete his deliberations over the weekend.
Whelan is looking for £2million compensation to allow Martinez to break the remaining 12 months of his contract, a factor which may yet come into the final reckoning.
Everton left-back Leighton Baines gave Martinez his seal of approval.
"It all sort of seems to be going on at the moment," said the England international, preparing for a friendly in Brazil at the weekend.
"Thankfully we are in camp so I am keeping it at the back of my mind for now and concentrating more on England.
"Roberto Martinez has been linked with the job. He is a terrific manager so we will just wait and see."
Everton captain Jagielka, also on international duty with Roy Hodgson's side, was more circumspect with his opinion of who the new manager would be.
"I will be delighted whoever comes in," he said.
"Everton is a massive club and hopefully we can get a fantastic manager."