The Hammers trio were all forced to pull out of the matchday squad for the 3-0 defeat at Manchester City on Wednesday just hours before kickoff after coming down with illness.
Moyes has confirmed that all three players have since taken part in light training on Friday and will be assessed ahead of welcoming the Red Devils to the London Stadium on Sunday, although one unnamed players has been sent home with illness as is therefore a doubt.
Speaking at a press conference on Friday, Moyes said: "[Rice, Soucek and Aguerd have] been in today. Today was the first time we've had the chance to assess them - they've done a little bit of training, so I'm hoping that they'll all be available for the weekend.
"One other player was sent home this morning and we've got another one who just said he didn't feel so good, so we've just got to hope that we can contain it.
"It looks as if it's more like a sort of sickness and the players are getting over it quite quickly."
Moyes has also confirmed that defensive duo Kurt Zouma and Vladimir Coufal will miss Sunday's game due to respective ankle and hamstring injuries, while Gianluca Scamacca remains sidelined after undergoing knee surgery last month.
Zouma missed the midweek defeat at Man City after twisting his ankle in the 4-3 loss at Crystal Palace last weekend, while Coufal sustained his injury in the second half against the Citizens.
"We're not going to give a timeline on Kurt, because we know he is a boy who does improve very quickly," said Moyes. "But he's also got an ankle injury so we've got no real date on that at the moment.
"Vlad won't be available for this game. He's hurt his hamstring and we're not sure exactly how bad it is. We're hoping it's not too bad."
A run of three successive Premier League defeats has prevented West Ham for climbing away from the relegation zone, with Moyes men sitting 15th in the table and just four points above the bottom three with four games remaining.
The Hammers have already faced Man United twice at Old Trafford this season, losing 1-0 in the league in October before suffering a 3-1 FA Cup quarter-final defeat in March - a game that they were leading with 15 minutes remaining - and Moyes believes his side were unfortunate to not come away with positive results in those previous meetings.
"We felt we could have got more when we played Manchester United earlier this season and there have been a few games where we could say the same thing," said the former Red Devils manager.
"We did a lot of good things at Old Trafford but we didn't come away with the results and if you look at recent seasons you think back to when Jesse Lingard scored a winner for them and I brought on Mark Noble in the 91st minute and he missed a penalty for an equaliser.
"A few of the games against Manchester United have been pretty close, so hopefully we can give them a good game and obviously it's really important."
Moyes added: "Manchester United are a really good team with terrific players. The manager (Erik ten Hag) has done a brilliant job with them this year, so overall they've got a good side.
"We had to play one of the best teams a few weeks ago in Arsenal [and got a result]. Now we've got to play another of the best teams in the Premier League in Manchester United, so we'll have to play very well.
"I'm looking forward to it, we need to play at our best. If we play at our best we'll give them a really good game and if we get the crowd behind us the way they did against Arsenal, it'll give us a great opportunity of getting a result."
The last time West Ham claimed maximum points on home soil against Man United was in September 2019 when goals from Andriy Yarmolenko and Aaron Creswell helped secure a 2-0 victory.