The 35-year-old striker scored 28 goals and provided 10 assists in 46 outings for the club during the 2016/17 campaign before suffering a knee injury, which ended his season prematurely.
However, the Swede, whose previous deal expired with United, has returned to Old Trafford as a free agent. He has been handed jersey number 10 by manager Jose Mourinho.
Ibrahimovic said: "I am back to finish what I started. It was always mine and the club's intention for me to stay. I cannot wait to get back out on that Old Trafford pitch, but I also know that I have to take my time to make sure that I am ready. I have been working hard and will continue to do so to make sure I am in the best possible condition for my return to the pitch."
Mourinho added: "We are delighted Zlatan is on the road to recovery and we are equally delighted to have his ambition and experience back with us. After his contribution last season he deserves our trust and we will be patient waiting for him to return. I have no doubt that he will be important in the second part of the season."