The Dutchman has been under pressure after his team’s poor start to the campaign despite extensive transfer spending in the close season.
Everton have won just twice in eight Premier League matches this term.
However, Koeman told the media on Wednesday — on the eve of their Europa League clash with French side Lyon — that he felt reassured following last Friday’s chat with Moshiri.
“We spoke about football,” said Koeman, who came to Everton last season after a successful spell at Southampton.
“There was not really a message but the feeling is that they (Everton’s board) are behind the team, they are behind the manager.
“Everybody knows in football that’s a nice thing but in football always, finally, it’s all about results. Until now it’s full, total support from the board, yes.”
Koeman, whose decision to persuade the board to pay £45 million ($59 million) for Gylfi Sigurdsson is yet to be repaid with performances from the Icelandic star on the pitch, joked when asked if he felt he was four matches away from a crisis.
Craig Shakespeare had remarked last weekend he felt managers were always four games away from coming under scrutiny. He lasted just one more match, a 1-1 draw with West Bromwich Albion, before being sacked by Leicester City on Tuesday after a six-game winless run.
“Maybe I’m in the crisis?” said Koeman.
“But everybody knows in football the manager’s job is a really difficult job because things change really fast. Most of the time the manager doesn’t get time to improve the team.
“It’s football, it’s hard to take for the managers but it’s part of the job.”
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