"He's not available," Ten Hag told a news conference on Wednesday. "The consequence after Arsenal. He has a muscle injury.
"No, I don't think [he will be out] too long, but I cannot tell how long. It's not really bad and we expect him to be back quite soon."
Rashford's rejuvenated form has raised questions as to whether he should be included in Gareth Southgate's England World Cup squad and Ten Hag believes he deserves the chance.
"It's quite clear [he can make England squad], sure, yes," the Dutchman said. "He's shown his great potential and quality."
The appearance of Cristiano Ronaldo has been highly anticipated by Sheriff supporters but the Portuguese forward has only started two matches in all competitions since the start of the Premier League season.
Ten Hag remained tight-lipped on whether Ronaldo would feature in Moldova.
"I'm sorry but the Moldovan fans will have to wait until tomorrow," he said.
It will be the first time United and Sheriff have met, and both had opposing starts to their Europa League campaign.
Ten Hag's side slumped to a 1-0 defeat against Real Sociedad while Sheriff picked up a 3-0 win over Omonia -- the United boss is not underestimating his opposition and is expecting a tough match.
"It's quite clear, they beat Real Madrid and Shakhtar, they are capable," he said.
"We have to make it our game but we are conscious of that fact. They are a serious and decent opponent and we have to play our best for the win."
Meanwhile, United goalkeeper David de Gea reiterated the significance of the match, despite believing his side should not be competing in the Europa League.
"I think it's a great competition," the Spain goalkeeper said. "We should be playing the Champions League, but this is the competition we play for, so we play for everything.
"I have great memories of this competition and tomorrow we have another big game against Sheriff."
De Gea, who is in his 12th season with the club, still claims United's spot for first-choice goalkeeper.
Dean Henderson, who left to join Nottingham Forest on loan in the summer, previously said he was "fuming" about his treatment at the club and their failure to deliver the promise of making him their No. 1.
But De Gea said that it is important to focus on the future of the club.
"Well, I'm just here to do my job, to try to better every day, so I don't think it's the time to talk about what happened last season," he said. "I think we should focus on the season now."