Midfield duo Gareth Barry and James McCarthy sustained ankle and hamstring problems respectively in the Merseyside club's 1-1 draw at Sunderland last time out.
Leighton Baines then pulled out of the England squad after suffering a hamstring injury in training during the two-week international break, while Darron Gibson complained of pain in his knee after playing for the Republic of Ireland against Scotland last Friday.
Steven Pienaar (knee) and Kevin Mirallas (hamstring) will also be assessed along with Seamus Coleman (Achilles wound), while Bryan Oviedo, Arouna Kone (both knee), Antolin Alcaraz (dislocated shoulder) and John Stones (ankle) remain sidelined.
Martinez is hopeful that his squad will not be significantly depleted when Everton attempt to stretch their unbeaten run to seven games in all competitions on Saturday.
"We're checking Leighton on a daily basis," said the Everton manager. "He's someone that heals really quickly.
"He had a grade one injury on Friday, that was with England. At the moment, it's difficult to know whether he's going to be involved on Saturday or not. It's touch and go.
"Normally it takes between 10 and 14 days - with Leighton you can't really rule him out. I've seen him recover from a grade one hamstring problem in eight days."
Martinez added: "James McCarthy picked up a grade one hamstring problem against Sunderland and we're assessing him. We'll see how he recovers.
"Darron Gibson hasn't been able to train yet. His knee suffered a little bit in the game against Scotland.
"Plus Seamus Coleman played with five stitches against Scotland, so we need to check the wound.
"Then you've got Steven Pienaar and Kevin Mirallas, they've been joining the group, and Gareth. I'm not too sure if Gareth will make the game, I'd be very, very surprised if he did."
Everton have slipped to 10th spot after drawing their last two top-flight matches and face a West Ham side revelling in occupying the heady heights of fourth place.
The London club have also drawn their last two games following a run of three consecutive wins and manager Sam Allardyce has played down talk of European qualification.
"We've made a good start in that direction but there is a long, long way still to go. I think it's a huge ask which is probably just ahead of us at the moment." said the 60-year-old.
Allardyce has concerns over key duo Stewart Downing (knee) and Diafra Sakho (back) ahead of the trip to Merseyside, but revealed that Andy Carroll is ready to start if needed despite not being fully fit.
West Ham have failed to win in their last 13 encounters with Everton - losing the last four - and their last victory came back in 2007 when Bobby Zamora scored the only goal of the game.
