There is "a lack of commitment" from both sides with regards to Moyes staying at West Ham, and Shakhtar Donetsk manager Fonseca is one of two managers mentioned as attracting West Ham's interest.
Moyes is credited with getting the best out of £25m forward Marko Arnautovic. "We'll wait and see when we meet up".
"I speak regularly with the board".
The former Everton, Manchester United, Real Sociedad and Sunderland boss was handed the job at the London Stadium earlier this year after Slaven Bilic was sacked and has guided the club to Premier League survival.
"I'd think you'd need to give a bit longer".
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"Look, whatever way it comes around, we have - and I say we because I mean my staff, Alan Irvine, (Stuart) Pearcey and Billy (McKinlay), they are all with me in the same situation - played well, improved from the start of the season no end".
"I'm still not sure the supporters have taken to him, but you can see - especially yesterday - there's a real base there".
When asked for his thoughts on taking three points against his former club for the first time in six attempts as a manager, he said: "I can only ever talk good of Everton and what they are like".
"What I'm being told is that if they part company with David Moyes, the kind of names they would be looking at to replace him would be Rafa Benitez and also Manuel Pellegrini".
It's reported that Moyes met with the club hierarchy on Monday to discuss his future.