Former Bayern Munich boss Niko Kovac watched managerless Everton's win over Chelsea from the Goodison Park directors' box on Saturday.
The 48-year-old Croat was not sat with Everton's owners for the 3-1 home victory, overseen by interim caretaker Duncan Ferguson and which lifted the club out of the bottom three.
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It is understood that Kovac has travelled to England on a 'fact-finding' mission about the English game, having left Bayern by mutual consent in November.
From Goodison, he was due to travel to the Etihad Stadium for the Manchester derby between City and United, before attending the Hawthorns on Sunday to watch West Brom against Swansea.
But his appearance has only heightened speculation among supporters that he is one of the names on a four-man shortlist of candidates to replace Marco Silva on Merseyside.
Also understood to be on that list is the Shanghai SIPG boss Vitor Pereira, who is due to begin negotiations with Everton over the weekend.
Ex-Toffees boss David Moyes is also said to be on the list and the preferred choice of chairman Bill Kenwright, while Napoli manager Carlo Ancelotti is another name to have emerged.
Kovac won a Bundesliga and German cup double as Bayern Munich boss last season, having taken over from Jupp Heynckes on a three-year contract.