
Both players came off before Virgil van Dijk scored an injury-time winner to earn Liverpool a last-gasp 2-1 win over Everton in the first-ever Merseyside derby at Hill Dickinson Stadium last weekend.
Branthwaite was stretchered off with a new hamstring injury and was never expected to feature against the relegation-threatened Hammers.
And, speaking at his press conference to preview the trip to the London Stadium, Moyes confirmed that the key centre-back will not play again this season.
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And the Blues boss also shared the bad news that the in-form Beto, whose equaliser against Liverpool was his fifth goal in six games, will also miss out this weekend.
Moyes, who will also be without long-term injury victim Jack Grealish, said: "Beto has concussion and will miss this week’s game.
"Jarrad will be out for the season, but then obviously we’ve got a long close season, so he’s got plenty of time to get himself fit and ready to start next season. But he won’t be fit to play again this season."
While Branthwaite faces another spell on the sidelines, Moyes said the England international, whose hopes of making Thomas Tuchel's World Cup squad are now over, was relieved the injury was not a recurrence of the one that meant he missed the first five months of the season and that required surgery in October.
Moyes said: "He’s much better because he's realised it isn’t an injury to the same hamstring operation area which he had - it's in a different spot.
"It’s like an hamstring injury that any player can get, and you might be out for six weeks, maybe a bit more, maybe a bit less, depending on where it is.
"You can see the shape of him - speed is a big part of his game. When you’re getting to that level there’s a bigger chance of getting hamstring injuries. But we feel for him because he played really well in the game."
Asked if another hamstring problem is a concern, Moyes continued: "I hope not. It’s a wee bit of an age thing - I don't want to say growing pains, because it's far from that, he’s a grown man now - but I think he’s still learning a bit more about himself, and what he does.
"I hope that’s the case. But ultimately he's had a couple of hamstrings now and we're obviously worried about it, yes."
Everton badly missed Branthwaite and Beto after they went off against Liverpool.
The result has left the Blues 10th in the Premier League table. Should they beat West Ham, they would move up to seventh and level on points with sixth-placed Brighton & Hove Albion going into the final four games of the campaign.