The Portugal international, 32, joined United in 2007, winning four Premier League titles and the Champions League in eight years at the club.
He went on to play for Fenerbahce, Valencia and Lazio before returning to Sporting for his second spell in July.
Nani has played 112 times for Portugal, with whom he won Euro 2016.
He leaves Sporting less than a month after helping them win the Portuguese league cup - the Taca da Liga.
"It is with a mixture of feelings that I leave the club of my heart," he wrote on Instagram.
"On the one hand, I'm excited to face another new challenge in my life, but on the other hand, I feel truly nostalgic about leaving a home that has given me so much. Sporting means everything to me.
"Sometimes life provides opportunities that are good for everyone. And this MLS tour is interesting for me and my family and very appealing to the club as well."
