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The 28-year-old spent a successful spell on loan at West Ham last season but Solskjaer said he sees his long-term future at Old Trafford.
“Jesse’s had a good start to the season,” Solskjaer told a news conference on Friday.
“He’s come on and scored a few goals for us that’s been very important for us. We want to prolong his contract, we see him as an important player for us.
“It’s up to him and maybe up to me to give him more playing time because he deserves to. Hopefully, we can get a deal sorted for Jesse because we really rate him.”
“International breaks are sometimes the time to cross your fingers and hope for the best because as we see with Raphael, he’s got an injury and is out for a few weeks now,” Solskjaer said.
“Fred and Edinson played a full game this morning, 1.30 a.m. (BST) kickoff. We’re not counting on them and they need time to rest and recover.
“They played well, not injured but hopefully they can be involved in midweek in the Champions League.
“Harry [Maguire] joined us on the grass this morning, that’s the first time he’s been on the grass.”
Despite potentially missing both first-choice centre-backs, Solskjaer said Marcus Rashford could make his return following shoulder surgery over the summer.
Rashford played 60 minutes in a behind-closed-doors game against Blackburn during the break and will make the trip to Leicester.
“He’s in the squad, if he’s gonna start or not, I can’t tell you now,” Solskjaer added.
“He’s worked really hard throughout the lay-off. He’s been really bright this week and had involvement behind closed doors last week, so he’s raring to go.”