Lanzini went off in the second half with a suspected hamstring injury, with Englishman Aaron Cresswell also leaving the pitch with injury as he limped off in the dying moments of the game -adding to the growing casualty list at the London Stadium.
Speaking after the game, Moyes gave an update on the two injuries, and claimed that he is hopeful that Lanzini's injury will not be serious, as quoted by Sky Sports.
"We've had a few injuries and we're just getting through. Behind the scenes we're just trying to nurse boys along. We hope Mano's [Lanzini] isn't serious. Cressy is okay, but it's more the Lanzini one that we're worried about.
If Lanzini is injured our board have to back moyes massively these next few weeks he's proved he's capable of doing well here back him get a couple of players in and push on ⚒— will earle (@willearle_) January 20, 2018
"I don't think you can suddenly say, okay we'll go out, see if someone else is out there waiting to be bought of the level of Lanzini, because I don't think you've got that. I think we will work with the players we've got, but we've got too many out injured at the moment."
The Hammers were on top for the majority of the game, and Moyes admitted that he was unhappy with the draw as the Scot rued his side's lack of killer instinct in front of goal.
Thought West Ham were great today and really enjoyable to watch - right up until the moment Lanzini went off and the entire team, shape and gameplan immediately feel apart. Could - and should - have won the game by three or four but now desperately in need of midfield creativity— Dan Silver (@dansilver_) January 20, 2018
"In the end, I wasn't unhappy with a point, but I have to say we missed some big chances which we could have got ourselves out of sight.
"I thought we played some of our best football this season, especially in the first half, but Bournemouth got strong in the second part of both halves, so in the end perhaps the draw was the right result.
"We won't to look there, but we're mindful that it won't take too much to slip down. It's another point to making sure we're in the right half of the league."
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West Ham remain 11th after the draw, and next up for the Hammers is a FA Cup fourth round clash against Wigan Athetic on Saturday, who coincidentally knocked out Bournemouth in the previous round of the cup competition.