The Armenia international is believed to be on the verge of completing a move to Old Trafford, with representatives from each club meeting in Manchester recently to sound out a deal.
Watzke, who promised ‘to assess Mkhitaryan’s situation’ earlier this week, has now claimed that United have increased their bid.
"We have to evaluate this as we start our pre-season training on Monday, and we will do that at the weekend," he told WAZ. "The situation has changed – Manchester United improved their offer.
"In situations like this you have to see things from different points of view and in the end we will make a decision,” he added.
"I have contacted the relevant people and we will decide at the weekend. We need the opinions of different people and after that the board of directors will decide."
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Although Watzke did not reveal a specific amount, several newspapers in Germany report that United have doubled their initial bid of £19.9million to £35.2m.
Mkhitaryan joined Dortmund from Shakhtar Donetsk in 2013 for £23.6m, having turned down advances from Liverpool.
After a difficult two years under former manager Jurgen Klopp, Mkhitaryan has blossomed under successor Thomas Tuchel.
The 27-year-old attacking midfielder, who has one year remaining on his contract at Signal Iduna Park, scored 23 goals and earned 32 assists in 51 appearances last season.
Mkhitaryan’s agent, Mino Raiola, recently described United as his client’s “dream club” and that the deal would be done “by any means”.
Mkhitaryan is in line to become Jose Mourinho's second signing since his appointment as the club's new manager in May, following the £30m capture of the defender Eric Bailly from Villarreal.
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