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The former Chelsea, Inter Milan and Real Madrid boss replaces Louis van Gaal, who was sacked on Monday, two days after United won the FA Cup.
Discussions, which started on Tuesday and were delayed by negotiations over Mourinho's image rights, finally finished on Thursday.
The 53-year-old Portuguese is expected to sign a three-year contract.
Mourinho's appointment has been backed by some former United players, including Peter Schmeichel, Rio Ferdinand and Steve Bruce.
"I have heard many say the 'Mourinho way' is not the Manchester United style of playing football," Ferdinand wrote on his Facebook page.
"However, the Chelsea team that Jose put together first time round, when he entered the Premier League, played with wingers and hurt teams with quick attacking play.
"This is a sign he knows how to get a team playing in a way Manchester United fans can look forward to seeing."
He added: "One thing I can tell Manchester United fans is that they have got themselves a true winner in all aspects. His CV shows success."
Zlatan Ibrahimovic, who worked with Mourinho at Inter Milan, could be on his way to Old Trafford, too.
The Sweden striker has left Paris Saint Germain and said he has had several offers from around the world.
"The future is written," said the 34-year-old. "I made my choice a long time ago. I just have to push the button."
Asked about a move to United, he responded: "Let's see what happens."
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Summarize: Jose Mourinho is expected to be officially confirmed as Manchester United's new manager on Friday.


Problem: Paul Hartley, 40, was sacked on Monday after seven straight defeats left the Dark Blues second bottom of the league.
And Rae, 47, expects there to be "plenty of takers" for the vacancy, adding that he would be interested.
"They made the decision on the back of what the ramifications would be if they get relegated," Rae told BBC Scotland.
"They'll need to get someone in quickly who knows the level, who is going to try and galvanise the boys going into the post-split.
"Paul's obviously done a decent job, he managed to get promotion out of the Championship, a top-six finish as well. He's had a good stint.
"It's been a tough year for him, having lost his two heavy hitters in the summer - Kane Hemmings and [Greg] Stewart.
"They need to focus on the short term now, survival is paramount for them."
Rae was manager at Dundee between 2006 and 2008 while the club were in the second tier, having been relegated in 2005. They were back in the Scottish Premier League for the 2012-13 season but were demoted again.
Former Alloa Athletic boss and one-time Dundee midfielder Hartley replaced John Brown at Dens Park in February 2014. He led the side to Championship title success, and a top-six place in their first season back in the rebranded Premiership, followed by an eighth-placed finish last term.
Goals and wins have been harder to come by for Dundee this season following the departures of forwards Hemmings and Stewart in the summer transfer window.
The Premiership splits in two for its final five rounds of fixtures - the first of which has Dundee scheduled to visit Motherwell on 29 April, with the Dark Blues five points above Inverness Caledonian Thistle.
The side finishing bottom will be relegated and the 11th-placed team will face one of the Championship's top sides in the Premiership play-off final.
Rae expects Hartley to "bounce back" and said of the current squad: "There's some decent players.
"It is a fabulous club, Dundee. There'll be plenty of takers [for the manager's job]. Of that I'm certain. You tend to find that applications for this are substantial - 50, 60, 70 applications for a job like Dundee.
"Alan [Stubbs] will obviously be one of the names [linked with the job] for what he did, having won the Scottish Cup last year with Hibs."
And, when asked if he would be interested in a return to Dens Park, former St Mirren boss Rae replied: "I'm not exactly sure what they're looking for but if it was an opportunity for me to get back into the game I would jump at the chance."
What was that article about?

A: Former Dundee manager Alex Rae says keeping the club in the Scottish Premiership is "paramount" as they seek to appoint a new boss.


Problem: Homes were evacuated in Lurgan after officers responded to a call reporting an unexploded device - but, as they searched, a bomb exploded.
Police believe the initial call was designed to lure officers to the area.
Supt David Moore said it "may well have been the case" that the bomb was detonated by people watching from a distance.
He said it would probably have been "much larger" than a pipe bomb and was "obviously designed to take out officers on foot".
"It's more than likely at this stage that this was an anti-personnel device," Supt Moore said.
"It bears all the hallmarks of violent dissident republican terrorists."
He said the area where the device exploded is "frequented by both younger children at play and older children sometimes guilty of creating something of a nuisance".
He said as a result it would be an area often patrolled by the police.
The security alert began following a call to a charity, during which the caller said a device had been fired at a police patrol but had failed to explode.
Officers found a suspicious object and evacuated families from homes in Victoria Street.
However, while police officers were searching the area a second device went off.  The explosion happened between 16:00 and 17:00 BST.
Police said on Sunday that the devices they recovered "appeared to have been planted" and they did not believe any device was fired at them.
Police officers were also attacked with petrol bombs and bricks during the security alert.
Northern Ireland Justice Minister David Ford said whoever was behind the attack "had a clear intention to kill police officers".
"They showed absolutely no regard for local residents, who could have been killed or injured," he said.
Upper Bann MLA John O'Dowd, from Sinn FÃ©in, also condemned those responsible, saying they had  "brought nothing but disruption to the local community".
"Those behind this alert are not representative of the people of Lurgan and should stop these pointless actions immediately," he added.
As a result of the security alert, the rail line between Lisburn and Portadown was closed.
Families have now returned to their homes in Victoria Street following the alert.
SDLP MLA Dolores Kelly described the attack as "not only an evil attempt to kill the PSNI but a total disregard for the people of Lurgan".
DUP councillor Carla Lockhart said the devices "were designed to murder police officers who were trying to serve their community".
Ulster Unionist councillor Colin McCusker said he felt nothing but "despair" with regard to the incident.
He said a small minority appeared "intent on disrupting the lives of the vast majority of people, not just in Lurgan, but the whole of Northern Ireland".
Police Federation Northern Ireland chairman Mark Lindsay said it was only the professionalism of those responding to the call that had meant tragedy was averted.
"The bravery of these officers can only be contrasted with the cowardice of those plotting such atrocities," he said.
What was that article about?

A:

assistant: A bomb which exploded  in County Armagh on Saturday was "absolutely designed to kill" police, the PSNI has said.