The Toffees currently lie two points adrift of fifth-placed Arsenal and a further four points behind fourth-placed Tottenham ahead of Sunday's visit to White Hart Lane.
Moyes' side also face daunting away trips to Arsenal, Chelsea and Merseyside rivals Liverpool before the end of the season and the Scot believes his team may have to replicate their best-ever points tally under him - 65 when they finished fifth in 2008.
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"With eight games to go last season we had 40 points, this time around we have 51 and if we want to have any chance of getting close to Europe I feel we will certainly have to get above the 60 points mark."
The Scot also revealed that this time of the season traditionally sees his side take a break ahead of the vital run-in but, due to a fixture pile-up, the Toffees will have to do without the chance to recuperate.
"In recent seasons I would have looked to have taken the team away to have a few days training in the sunshine, but the way the fixtures have worked I have not found it possible this year," added Moyes.
"At this time of the season we try to keep the training short and sharp as much as we can. It’s a pattern we've followed in previous years and it has tended to work for us."