Solskjaer is desperate to bolster his defensive ranks after a particularly shoddy year at the back.
United conceded 54 league goals – their worst tally since 1978/79 – and they will now be playing in the Europa League for the second time in three years.
The Red Devils have already got two deals over the line but time is fast running out in the transfer window – which shuts in two weeks.
Daniel James and Aaron Wan-Bissaka have been signed but there are pursuits ongoing for the likes of Maguire, Sean Longstaff and Bruno Fernandes.
But there are problems being met with the former, with Leicester demanding a whopping £80m.
That is £10m more than what United have offered, who have so far been reluctant to break the world-record fee for a centre-back.
That figure currently stands at £75m after Liverpool forked out for Virgil van Dijk.
Nevertheless, Sky Sports report that Solskjaer is hellbent on the Red Devils getting the deal over the lines.
He has told executive vice-chairman Woodward to do everything he can to confirm the signing and it is now in the United chief’s hands.
Woodward decided to stay in England instead of flying out with the squad on their pre-season tours of Australia and Asia.
The former banker decided not to travel as he continues to work on transfers.
Maguire would love the chance to move to United and wants to play European football.