Inter have an option to extend the deal for Young for a further season and he joins up with former Old Trafford team mates Romelu Lukaku and Alexis Sanchez. British media reports said Inter had paid a fee of 1.5 million euros ($1.66 million) to United.
Young, 34, who has converted to fullback after beginning his career as a winger, was out of contract at United at the end of this season.
United manager Ole Gunnar Solskjaer said on Friday that the club was not going to stand in the way of allowing Young to move to the Italian giants.
"Ashley's mindset was that he wanted to try a new adventure and he's approaching 35 now. He's done a fantastic job and he's served this club fantastically," Solskjaer said.
"I think it was an opportunity for him that we had to say yes to. We got a transfer fee for someone who's been so loyal to the club."
Young began his career with Watford before moving to Aston Villa in 2007 and joined United four years later. He has been capped 39 times by England with his last appearance coming in the World Cup semi-final defeat by Croatia in 2018. ($1 = 0.9018 euros) (Reporting by Simon Evans Editing by Christian Radnedge)