The 21-year-old would not have played for Man United before the end of the month regardless of whether his deal was announced in the early stages of July, though, as he is still on holiday following his involvement in this summer's European Championship.
Solskjaer's side recorded a 2-1 victory over Derby in their pre-season opener last weekend and will continue their preparations against Queens Park Rangers on Saturday afternoon.
The Red Devils will then finish July with a clash against Preston North End before rounding off their preparations for the new campaign with a match against Everton on August 7.
In theory, Sancho, who has already undergone a medical and agreed personal terms with his new club, should be available to make his first appearance for Man United against Everton, with an official announcement on his signing likely to arrive before the end of the month.
The attacker's competitive debut for the 20-time English champions could then come one week later against Leeds United at Old Trafford.
Sancho, who scored 16 times and registered 20 assists in 38 outings for Dortmund last term, will be behind the majority of his new teammates in terms of preparations for the new campaign, while he only actually played 126 minutes at Euro 2020.
As a result, the former Manchester City youngster will need time to get up to speed with his new club, but there is every chance that he will feature against Everton on August 7 before pulling on the famous strip for the first time in a competitive contest against Leeds in Man United's Premier League opener.
On paper, Solskjaer's side have a kind start to the season, and it will be fascinating to see how Sancho performs for his new club following an excellent period at Dortmund.
Potential first-team debut: August 7 vs. Everton (friendly)
Potential competitive debut: August 14 vs. Leeds United (Premier League) body check tags ::