The Norway international is widely expected to leave BVB during this summer's transfer window, when a £63m release clause in his contract will become active.
Man United are long-term admirers of the 21-year-old and were expected to be in the fight to secure his services ahead of the 2022-23 campaign.
However, according to ESPN, the 20-time English champions have dropped out of the race and are now looking at alternative targets in the final third of the field.
Manchester City and Chelsea have also been linked with the youngster, but Los Blancos are seemingly set to win the race for his signature, potentially signing both Haaland and Kylian Mbappe this summer.
The Dortmund striker has again been in excellent form at club level during the 2021-22 campaign, scoring 22 times and registering six assists in 19 appearances in all competitions.