The Everton boss watched as his side were comfortably beaten by Manchester United on Sunday afternoon and they have now dropped to sixth in the table, six points behind Spurs who currently occupy the final Champions League spot in fourth.
Moyes is worried that he has not got the chance to rest his star players and is calling on them week in, week out for a massive effort.
"We are asking people like Leon Osman and Steven Pienaar, who we rely on heavily, and Leighton Baines, to put a big shift in and do it for us every week," he said.
"Maybe the numbers will allow us to have a good half season, but not a good full one. I hope I am wrong.
"We do tend to finish strongly. This time we have had a really good opening, so I hope we can do it again.
"But we are asking a small number of players to do it for a full season."