Rice was not included in the Liverpool squad after falling ill on Thursday while Bowen had to be helped into the Anfield tunnel after sustaining a second-half injury.
The winger, who has been in sensational form this season and is in contention for a first England call-up this month, will undergo a review before Thursday.
West Ham are due to travel to Sevilla for the first leg of the Europa League Round of 16 in the coming days, their biggest game of the season.
However, the West Ham manager was unable to give any clear updates on the two players after the narrow defeat at Liverpool.
“They seem to think the tackle of [Andy] Robertson, around his heel,” Moyes said.
“I don’t know where but we’ll have it scanned. Jarrod has been exceptional for us, so the last thing we want is for him to be injured.
“It couldn’t have been anything, but in the same breath, I didn’t like the way he wasn’t able to put weight on his foot, which worries me.
“He played very well for us, so it would be a blow if we went into the Sevilla game without him.
“Declan had an illness which occurred on Thursday and Friday. We hope it won’t be too serious, but I can’t say at the moment.
West Ham squandered a number of chances to come away with at least a point from Anfield, with Pablo Fornals, Manuel Lanzini and Michail Antonio all wasting good chances.
“They started off really well but we grew,” Moyes said. “We were very lucky when Pablo [Fornals] was over but he didn’t. And late, if a team was going to score a goal, it was us.
“We have to be clinical, the quality of the last finish and the cross that has been lacking.
“I thought Michail Antonio was outstanding today but we still need that bit of class at the end that we’ve been missing for the past few weeks.”