The deal is subject to obtaining a work permit. The 23-year-old striker joins from Olympique Lyonnais and has previously had spells at Paris Saint-Germain and Fortuna Hjørring.
Bruun currently has 19 caps for Denmark and has scored 14 goals for her country. She was also recently voted as Denmark’s Player of the Year.
Signe Bruun said: “It’s amazing to be here, I’m really happy to be joining Manchester United. It is a big club with huge history, I have followed the women’s project for a while and have seen the progression of the team, and I think this move was a perfect match for me.