The battle to claim a coveted top-four berth has crystallised into discussion over who will get squeezed out from the London trio of Tottenham, Chelsea and Arsenal.
Presently six points behind with a game in hand, Everton would move to within striking distance if they were able to overcome Spurs at White Hart Lane on Sunday.
It is a situation Moyes relishes. But he is not ready to blow his side's cover just yet.
"At the moment I am happy to stay under the radar," said Moyes. "If we are in the middle of April and people start to talk us up, that will do me nicely because it will mean we are coming into the final furlong with a chance.
"If we could somehow get our head in front that would be great."
With visits to Arsenal and Chelsea still ahead of them, Moyes concedes his side will have to do it the hard way.
However, games of such significance have not been in plentiful enough supply at Everton down the years. And Moyes is happy to take them on.
"High-pressure games are what we have been craving," he said. "Partly because we have not been good enough, or won the games that mattered, we have not had as many as we would like.
"I do accept it is much harder for us because we have got all those teams to play away from home, so of all those who have a chance of getting into Europe, we are certainly the underdogs."
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