The Red Devils' interest in the Englishman has been a theme of the summer and the report says that talks have now begun at the top level of the two clubs over a move.
Although United recognise they are operating in an inflated market and there is a premium on young British players, there is some concern that there is nearly an additional tax on players they are interested in signing.
Gone are the days of Galactico signings at Manchester United.
Solskjaer stressed on Wednesday that United will not be held to ransom and want "the right players at the right prices", but they are likely to have to make Harry Maguire the world's most expensive defender if they are to prise him away from Leicester City, which has become the subject of their focus of late.
Recent reports claimed that Manchester United will have to spend as much as £50m to sign Longstaff.
The Old Trafford outfit shipped 54 goals in 38 games in the Premier League to raise questions about their defence under Solskjaer.
Keeping Longstaff would be a rare chink of life in a summer which has been nothing short of dire, but if United come forward with the kind of money they threw at Aaron Wan-Bissaka, it would not be a surprise to see Newcastle cave.
Manchester United have also snapped up England Under-21 worldwide Aaron Wan-Bissaka in a £50m deal from Premier League rivals Crystal Palace.
Solskjaer was appointed as Jose Mourinho's replacement in December before the Norwegian was handed the reins on a full-time basis in March.