The 34-year-old Uruguay international moved to Old Trafford on a free transfer at the start of the season, signing a one-year contract.
But the impact of the former Palermo, Napoli and Paris Saint Germain forward has been immense, prompting Manchester United to offer him fresh terms.
Cavani has scored 15 goals in 35 appearances across all competitions, helping Manchester United to second on the Premier League table and a place in the Europa League final, where they will take on Villarreal.
In a statement, Cavani said: "Over the season, I have developed a great affection for the club and everything that it represents.
"I feel a deep bond with my teammates and the staff who work behind the scenes here. They give me extra motivation every day and I know that together we can achieve special things.
"From the very first moment that I arrived I felt the confidence of the manager, as a player this belief gives you the perfect opportunity to play your best football and I want to thank him for that.
"I have been touched with how much the supporters have wanted me to stay, and I will give everything to bring them happiness and joy with my performances on the field.
"I’ve not yet been able to play in front of the Old Trafford crowd and that is something that I cannot wait to do.
