Allardyce watched from the stands at Goodison Park, as Everton picked up a 4-0 win over Moyes’ men.
Allardyce will take charge of his first match in charge of Everton this weekend, when they take on Huddersfield Town.
And the former England boss has already been told how great a club he is taking over at by Moyes.
“I saw Sam Allardyce for 10 minutes in the office after the game on Wednesday,” Moyes said.
“I told him he was going to a good club and I think it suits Sam.
“The Everton job was one of only a few which would have tempted him back. He can live at home in the North-East and I am sure that also played a part in his decision.”
Moyes was in charge at Everton for 11 years, and it was the last job he was truly successful in.
The Scottish boss has failed at Manchester United, Real Sociedad and Sunderland since - and the signs are worrying for him at West Ham.
Moyes has failed to win any of his first three games in charge at the Hammers, with the London side now sat in the relegation zone.
Everton, meanwhile, are currently in 13th place and now have some momentum behind them, ahead of Allardyce’s first game in charge.