The Catalan club are in action against Sevilla in the Uefa Super Cup in Tbilisi on Tuesday night and Pedro is expected to feature after Neymar was ruled out due to illness.
Barca are also keen on signing Celta Vigo forward Nolito as a possible replacement for the winger, but will only push through a deal once Pedro's move to Old Trafford has been finalised.
Pedro is keen to leave as he seeks more first-team opportunities and there is a gentleman's agreement allowing the player to depart for a cut-price of €30 million.
Neymar's mumps, which will also see him miss the Spanish Supercopa double-header against Athletic Club on August 14 and 17, could hold up the move slightly, but Luis Enrique admitted on Monday: "Pedro's future will be resolved shortly because there is not long to go until the transfer market closes."
Speaking at the pre-match press conference ahead of the game against Sevilla, he added: "The season is about to start and it's time to focus. Every competition is a challenge and we will have to see what we do and what our rival does to see who deserves the title."
Barca won the treble last season and will now look to replicate the six titles won in 2009 by Pep Guardiola's great Blaugrana side as they compete in the Spanish and European Super Cups and the Club World Cup at the end of the year.
"Our work has gone to plan and we arrive in a good moment," Luis Enrique said.
In Neymar's absence, the Asturian coach lined up his side with Pedro in attack in Monday's training session - alongside Lionel Messi and Luis Suarez.
And on the Uruguayan, he said: "Suarez had a spectacular year. He was very important for us and few could have matched him. He will have the same role again this season."