The Old Trafford club made a £72m bid yesterday but they are prepared to go to £80m to get their man, says The Sun.
Personal terms based on a five-year deal worth £350,000-a-week have already been agreed.
Despite reports of wrangling, United have long been comfortable about the transfer going through smoothly after a year-long pursuit of the England international, 21.
This time last year they embarked on a summer of bartering before refusing to meet an asking price of £110m from the Germans.
United also spoke to Erling Haaland before the Euros, but the Dortmund striker has his heart set on a move to Real Madrid.