Rashford picked up the injury after scoring twice in United's win over Arsenal on September 4th and missed their Europa League opener, a home defeat to Real Sociedad last week.
"He is not available. It is still the (same) consequence after Arsenal. He has a muscle injury," Ten Hag told reporters on Wednesday before Sheriff host United in Moldova.
"I don't think (he'll be out) for too long, but I cannot tell how long. It's not really bad and we expect him to be back quite soon."
Rashford, who has not featured for England since Euro 2020, had been tipped for a return to the national team after his impressive start to the season, having scored three goals and made two assists in six Premier League games.
He is now an injury doubt for England's Nations League games this month, however, but Ten Hag backed him to make Gareth Southgate's World Cup squad for November and December.
"That's obvious and quite clear, yes," Ten Hag said. "I think this season he has had a great comeback and he has showed his great potential and quality."
Rashford, Donny van de Beek (25) and Aaron Wan-Bissaka (24) did not train on Wednesday while Anthony Martial (26) continues to be sidelined after sustaining an Achilles injury following the win over Liverpool last month.
Moldovan champions Sheriff, who beat Omonia Nicosia 3-0 in their opening Europa League match, shocked Real Madrid at the Santiago Bernabeu in their debut Champions League campaign last season.
"It's quite clear, they beat Real Madrid and Shakhtar (Donetsk), they are capable," Ten Hag 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.
"When you lose the first game, you have to win the second, so we know what our task is."