Lingard has been a key part of United's first team for the last four seasons.
Last term, he made 27 appearances in the Premier League, netting four goals and providing two assists, some way down on his eight goals and five assists in 2017-18.
Now 26, the Englishman should be in the peak years of his career, and Solskjaer has made it clear he is expecting more from Lingard in the new season, per Jack McGraghan of Goal:
"Jesse has proved before with his big-game goals that he’s valuable. But of course I’m going to demand more of him. I think he will be the first one to know that his last season was not up to standard.
"What we have discussed can be between the two of us, but I expect more from him and he knows what I want. Jesse knows what it is to be a Manchester United player. At times maybe he’s made one or two bad decisions but he’s made more good decisions than bad. He’s a top boy."
He looked particularly good against the Nerazzurri in Singapore:
United eventually beat Inter 1-0 thanks to Mason Greenwood's 76th-minute goal to maintain their perfect record in pre-season, which also included a 2-0 win over Glory and 4-0 defeat of Leeds.
Greenwood could have a role to play for United in the new term after some impressive recent performances:
The 17-year-old Greenwood, Lingard and Marcus Rashford are all graduates of United's academy, and the trio could theoretically make up the Red Devils' attack in 2018-19, but there is plenty of competition.
Alexis Sanchez, Anthony Martial, and new signing Daniel James will also be in Solskjaer's plans for the opening match of the Premier League season on August 11, when United host Chelsea at Old Trafford.