United announced they had agreed a £34 million deal with Real Madrid for the defender on July 27 but 17 days later are yet to formally announce his arrival.
Varane is in Manchester after spending time in isolation but, according to Solskjaer, he has not been able to train with his new teammates.
The deadline to register players with the Premier League for the opening round of fixtures passed at 12 p.m. (7 a.m. ET) on Friday.
"He has not trained with us," Solskjaer told a news conference on Friday. "Things happen and things take time and unfortunately the isolation has delayed these things. It's not 100% done and I can say he probably won't be [available against Leeds] because he has not trained with us.
"It always happens that there are little things. The big hold up was the isolation and quarantine. He has been in Manchester now and hopefully it'll be done soon."
Sancho did not take part in the training camp in St. Andrews but returned to training on Monday.
"Jadon is going to be involved, definitely," Solskjaer said.
"He has had a good week. Unfortunately he was ill after his holiday and we lost out on a few days of training with him but he has gelled really well with the group and he is looked sharper than I expected. I can't tell you if he will start."
Bruno Fernandes has said he is excited about United's new summer signings but that it can't all be on them to win the club trophies.
"All the players the club hires certainly have a lot to add to the squad, but we can't expect them to come in and change the whole club, or to be the ones responsible for making the team better," he told ESPN.
"We know these are two very high-level players and we expect a lot from them, but the pressure we put on them is the same pressure we put on those of us who are already here. Everyone has to be under the same pressure.
"The responsibility grows because we want trophies, and we know that by hiring players of this quality we are closer to achieving this. At the same time, we cannot leave all the responsibility rests on their shoulders, and if for some reason we fail to hit our goals, it is not only their responsibility but also on the whole team."
United lost their first game of last season against Crystal Palace and Solskjaer is keen for his team to make a faster start this time around.
Old Trafford will have a capacity crowd for the first time since March 2020 after COVID-19 restrictions were eased in the UK.
"We've had a good preseason and the last couple of weeks have been really good with everyone back in," he said. "We're still missing a few players through injury and COVID-19 but we're still confident.
"This club is about trophies, football and my expectations are we keep developing our style and play good football.
"Hopefully when we get to March, April and May and we are there or thereabouts. The team is hungry, we've done good business. I feel confident going into the new season but the first couple of games are an unknown."