The Hammers' season appears to be drifting to a conclusion and the draw with Newcastle last Saturday was uninspiring, but Allardyce is looking for his side to enjoy the remaining games of the campaign.
Allardyce is expecting a stern challenge at Goodison Park but wants the east Londoners to put in a good performance as they search for only a fourth away win of the season.
"Everton is no easy place to go but we've got to try to go up there and get some points," he said.
"These games are about enjoying it, really.
"We can go out with no pressure and enjoy it and try to finish as high as possible. That's why we work so hard each year - to get out of relegation and look up the table."