While Dortmund are seemingly open to a big-money sale, the Bundesliga outfit would prefer any transfer to be concluded by August 10.
However, according to the Manchester Evening News, United do not feel in a rush to get a deal over the line.
With the England international having just two years left on his contract, the Premier League giants allegedly believe that there will be no issues with having a bid accepted at a later date.
Although there have been conflicting reports with regards to a fee, Dortmund are expected to demand in excess of £100m, with the inclusion of add-ons.
United are said to favour a proposal which would see them pay a large portion of the fee over a number of years.