England centre-back Maguire is reportedly set to become the most expensive defender in football history, with an £80 million ($141.4 million) price tag that would surpass the £75 million ($132.5 million) Liverpool paid Southampton for Virgil van Dijk in January 2018.
Speaking after Sunday's (AEST) 2-2 International Champions Cup draw against AC Milan in Cardiff, where Wales international Daniel James dispatched the winning spot-kick as United prevailed 5-4 on penalties, Solskjaer said the Maguire deal could be done "very soon" and would not rule out picking the player for next weekend's Premier League opener at home to Chelsea.
"Hopefully the last little details will be in place so we can announce something very soon," he said.
"Let's get the last details over the line [before discussing Maguire playing against Chelsea] but, yeah.
"He's played a few games over the summer. We'll have to assess him if and when."
United was without Paul Pogba at Principality Stadium, the midfielder's withdrawal with a back spasm viewed suspiciously in some quarters amid renewed links to Real Madrid.
Solskjaer insisted he has "no doubts" Pogba wants to be at United this season and expects him to be fit to take on Frank Lampard's side.
"He will be back next week training," he said. "It's not an injury, just some pain.
"I wasn't going to risk anything this weekend. He should be okay for Chelsea."
Romelu Lukaku did not travel with the United squad and Solskjaer said there was "no update" on the striker's future amid reports of a swap deal including Juventus forward Paulo Dybala.
Lukaku courted controversy by publishing sprinting stats from United training sessions on social media on Saturday, a post that was subsequently deleted.
"I've spoken to Rom, yeah, and that's case closed," Solskjaer added.