Evra signed a one-year contract extension with United eight weeks ago, but the defender has since been given permission to leave Old Trafford.
The 33-year-old's move to Turin had been put in doubt following the resignation of manager Antonio Conte, but his replacement Massimiliano Allegri was keen to continue the club's pursuit of the player.
Evra, who has been at United for eight years, is on holiday at the moment, but Press Association Sport understands he could be announced as a Juve player on Monday.
Evra will not be the last player to leave Old Trafford this summer.
When asked whether there would be any player departures, executive vice-chairman told MUTV on Sunday: ''Yes, I think so.
''The manager makes that decision and a few players have already gone and there may well be a few others that go.
''It's also worth commenting that we're not in Europe this year (so) there's less games, maybe the manager takes the view that he wants a tighter squad to bed in.
''If you do the analysis over last year, Liverpool performed extremely well with a smaller number of players but that's a decision for the manager.''
Woodward insists he will give his full backing to new manager Louis van Gaal in his quest to improve the squad following their seventh-placed finish last term.
The Dutchman has spent upwards of £50million on Luke Shaw and Ander Herrera.
He has also been linked with Arturo Vidal, Daley Blind, Kevin Strootman and Thomas Vermaelen.
Woodward says there is plenty of cash available to Van Gaal if he decides he needs to strengthen further.
''There is no fixed budget
Financially we are extremely strong, we have funds available,'' Woodward said in an interview filmed on the club's pre-season tour of the United States.
''Louis is the boss and is assessing what's going on but we have been in dialogue for some time about targets so there is a lot of work going on behind the scenes..
we're continuing to move forward on some of those targets..
so watch this space.
''The time he (Van Gaal) spends here (in America)as well, he'll be able to assess more what he's got and continue to tweak the decisions if we need to.
''We want to do what it takes to win the title.''
Woodward has become tired of seeing targets slip through his fingers.
"I want to stop seeing the best players going to other clubs in Spain, I want us to stand out as the best club in the world," he said.
"We'll do absolutely everything we can to achieve that."
Van Gaal put his 25-man squad through a double training session on Monday at the StubHub Center.
United will kick off their pre-season campaign with a match against the Los Angeles Galaxy on Wednesday, before then travelling to Denver, Washington and Michigan to play Roma, Inter Milan and Real Madrid as part of the International Champions Cup
Source : PA
Source: PA