Woodward watched the 28-year-old in action during Barcelona's 1-1 Spanish Super Cup second-leg draw with Athletic Bilbao last night.
And the United chief is holding talks with Barcelona today, despite Chelsea and Manchester City's interest slowing down the move.
GETTY
Barcelona star Pedro in Spanish Super Cup action last night
"Ed Woodward, the Manchester United chief executive, has been in Barcelona today, speaking to the club and the word is that a deal will be done this week one way or another," La Liga expert Guillem Balague told Sky Sports.
"I understand a deal between the player and United is already in place and all that is left is for United to put the €30m (£21.3m) on the table. They are just negotiating how they are going to pay it, with installments or an amount up-front.
"They are very close. I'm not sure whether Barcelona will try one last time with Pedro and say 'don't go', but it seems his mind is already made up."
We use your sign-up to provide content in ways you've consented to and to improve our understanding of you. This may include adverts from us and 3rd parties based on our understanding. You can unsubscribe at any time. Read our Privacy Policy