Moyes takes his fourth-placed team to Newcastle United on Saturday when a victory would push the Hammers into third spot.
"Chelsea, Liverpool, Tottenham, Everton are all teams that can touch us. We have seven to go I have my targets and the players know what we need to do.
The 22-year-old picked up the issue late in England's 2-1 World Cup qualifying win over Poland at Wembley last Wednesday but managed to finish the game.
Rice was absent for West Ham's 3-2 win at Wolverhampton Wanderers on Monday, the first league game he has missed all season, and local media reported he could be out for a month.
West Brom, second-bottom of the table, moved seven points from the safety zone while Chelsea stayed fourth but could be overtaken by West Ham United and Tottenham Hotspur this weekend.
"There's no need to concede five goals when you have our quality," Tuchel told BBC Sport after Chelsea's worst home defeat in nearly 10 years.
"Not really. I think they have a really strong squad and they have a great manager and coaching staff," Arteta told a news conference.
"He gets the group together, they're working really hard for each other, they've recruited really well and you can start to see the things that David has done."
Moyes, whose side are seventh with 45 points after 26 games, said the club do not require Champions League qualification to retain Rice, who is under contract to 2024.
City took the lead in the 30th minute with central defender Ruben Dias heading in a Kevin De Bruyne cross, but the Hammers drew level through Michail Antonio two minutes before the interval.
Pep Guardiola's side secured their 14th consecutive league win, however, when Dias' fellow defender John Stones slotted home after being picked out by a Riyad Mahrez pull back.
After the next five or six games, if we get down to single digits games to go, then I could probably start to see what could happen.
West Ham have not finished above Tottenham since 2008, though Moyes said he was not reading too much into their opponents' run of four defeats in their last five league games.
The 27-year-old Argentine had previously apologised after being pictured at a Christmas party along with Tottenham Hotspur's Erik Lamela, Giovani Lo Celso and Sergio Reguilon.
I thought a lot about my actions and I know I cannot change what happened but I've spoken to the club and the manager and want something good to come from it.