Rashford could not finish the 3-1 defeat to Paris Saint-Germain on Wednesday because of a shoulder problem -- the same issue which ruled him out of the last international break.
- Insider Notebook: Man United give Solskjaer backing
- Stream LIVE games and replays on ESPN+ (U.S. only)
But the striker was able to train on Friday morning ahead of the trip to West Ham and Solskjaer is hoping he will be able to feature at the London Stadium.
"Marcus has been training this morning, not fully but he joined in with the group," Solskjaer told a news conference on Friday. "His shoulder stopped him finishing the game [against PSG] so he's been getting treatment and hopefully he'll be part of the squad that travels. We'll see after training."
Solskjaer will go head-to-head with one of his predecessors at Old Trafford, David Moyes, who has led West Ham to three straight wins in the Premier League to rise to fifth in the table.
Moyes only got 10 months at Old Trafford after replacing Sir Alex Ferguson in 2013 and was sacked once it was mathematically impossible for United to qualify for the Champions League.
"I haven't sat down and had discussions with David, no," Solskjaer said. "When you're here it's the biggest club in the world and a different animal to anything else.
"I really admire David and you see the way West Ham perform and the results he's getting, it shows his qualities."
Victory for United at the London Stadium -- which will host fans for the first time since March -- would move Solskjaer's team into the top four for the first time this season. It would also leave United just two points behind leaders Tottenham Hotspur and Liverpool, who both play on Sunday.
"We know we've got a game less and hope we get that back at some point as well," Solskjaer said. "With preseason being as short as it was, to move closer to the top would be a great achievement at this point and we've shown some great consistency away from home lately and we hope we can continue that."