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He scored in a 2-0 victory over Manchester City earlier this month and many believe he is a key asset to the Red Devils attack.
The 25-year-old has now revealed Solskjaer thinks he is the club's best finisher - behind Mason Greenwood.
Shaw explained: "You should ask the manager he normally says me.
"I understand people will say I don't score many goals but he's actually said to me I was one of them. I don't know if he was being nice or not."
Solskjaer has since confirmed it and named highly-rated Greenwood as United's best pure finisher.
"I've said that to Luke yeah," Solskjaer admitted. "He is behind Mason Greenwood, Luke has a very good finishing technique when he gets in that position so I encourage him to get up there.
"That was my aim to get him higher up the pitch, getting the end product."
Shaw's form this season has seen him earn a recall to the England squad and he assisted Harry Kane in Sunday's 2-0 win over Albania.
It was his first appearance since 2018 and Shaw has admitted he feels he let manager Gareth Southgate down in recent years.
"It was just pulling out when I got to the camp," he said per BBC Sport.
"Around that time, I was maybe not in the best condition but I think over the last two years I've thought about it so much - that was my biggest regret.
"And yeah, of course, letting Gareth (Southgate) down. I tried to keep in touch with him just to let him know that things had changed.
"He's said he's picked me on merit and how I've been performing.
- Lost his place in the England squad- United Faithfuls (@UtdFaithfuls) March 28, 2021
- Suffered a horrific leg break which kept him out for a year
- Dragged publicly by Mourinho
- Now back to his best, back in the England squad and assisting.
No matter the club you support, you've got to be happy for Luke Shaw. pic.twitter.com/evuoVfVJbh
"I think playing for your country is a massive privilege and an honour and I think at times I took advantage of that.
"I didn't really think about it too much but of course now I just want to forget the mistakes I've made in the past, and Gareth's done that, so we're just focusing on the future and obviously what's happening now."