Bowen has once again been a standout performer for the Hammers so far this season, scoring 14 goals and providing five assists in 26 appearances across all competitions.
Eleven of those strikes have been netted in 19 Premier League games this term, with only Erling Haaland, Mohamed Salah (both 14), Son Heung-min and Dominic Solanke (both 12) having scored more at the halfway stage.
However, Bowen picked up an ankle injury in the closing stages of West Ham's 1-1 draw with Bristol City in the FA Cup third round just under two weeks ago, and he was forced to miss the Hammers' 1-0 defeat in a replay against the Robins on Tuesday.
Initial fears emerged that Bowen could be facing a lengthy period on the sidelines, but Moyes has confirmed that the England international has made progress in his recovery.
Speaking at a pre-match press conference on Friday ahead of Sunday's trip to Bramall Lane, Moyes said: "Jarrod is doing better.
"Will he be available? I won't be giving that information out, but he's certainly doing better than we thought."
West Ham will definitely be without Lucas Paqueta, who is nursing a calf injury, and Moyes has admitted that he has "no idea" when the playmaker will return to first-team action.
Paqueta is understood to have seen a specialist to discover the severity of his injury, and initial reports have suggested that the Brazilian could be sidelined for up to two months.
Meanwhile, Michail Antonio remains out with a knee injury, while Pablo Fornals and Edson Alvarez - who missed Tuesday's defeat at Bristol City - will be assessed ahead of kickoff.
"They're okay," Moyes added. "We'll see how they are those two and Jarrod we'll need to make late calls tomorrow and see if they're able to travel or not."
West Ham enter Sunday's contest sitting sixth in the Premier League table, six points behind the top four, after accumulating 10 points from their last four top-flight fixtures.
Moyes has talked up the qualities of Sheffield United boss Chris Wilder ahead of the trip to Bramall Lane, saying: "Chris Wilder has always done a really good job.
"I think he's won the Manager of the Year a couple of times for the work he's done at Sheffield United. He has a got a great history there and his teams have always been really hard to play against.
"Bramall Lane always has a great atmosphere so we know it would be a tough game. We're looking forward to playing the game. We need to travel to Sheffield but we're looking forward to it."
West Ham will be confident of success against relegation-threatened Sheffield United as they have won each of their last three meetings with the Blades by an aggregate score of 6-0, including a 2-0 victory at the London Stadium earlier this season.