But Monday's meeting, set up when Moyes was still manager, proves that Sullivan is certainly interested in making another managerial change, with Fonseca the first through the door for talks.
"What I'm being told is that if, if, they part company with David Moyes, the kind of names they would be looking at to replace Moyes would be Rafa Benitez at Newcastle United and also Manuel Pellegrini, who of course used to manage Manchester City, won the title with Manchester City, and is now working in China".
Dean Ashton has told talkSPORT he expects David Moyes to leave West Ham now the Premier League season is over.
I also understand that Sullivan already knows who he wants and Benitez, a man who he has nearly landed before, is his preferred candidate.
"I have another year of my contract and we are trying to do our best to do what we have to do to go forward".
"I'm still not sure the supporters have taken to him, but you can see - especially yesterday - there's a real base there".
"I hope very much that Rafa will remain at Newcastle United".
Everton boss Sam Allardyce responded with the introduction of midfielder Davy Klaassen at the break, but it was West Ham who doubled their lead with a fierce shot from Marko Arnautovic.
Former Manchester City supervisor Manuel Pellegrini can be on a shortlist of candidates however Fonseca maybe has the benefit of being a youthful coach with a rising repute.