The 34-year-old will undergo a medical at the Humanitas private hospital in Rozzano, close to Milan, as well as being given a tour of Inter's Angelo Moratti Sports Centre training base in Appiano Gentile.
England full back Young has made 18 appearances for Ole Gunnar Solskjaer's United this season, but started only 10 Premier League matches.
The United club captain was out of contract in the summer, but Inter boss Antonio Conte was keen to tie up the deal during this month's transfer window.
Young joined United from Aston Villa in 2011 and went on to make 261 appearances, scoring 19 goals.
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It is understood the Premier League veteran reacted angrily to United not wanting to let him leave, especially as he feels he is not getting enough first-team football.
Inter are second in Serie A, two points behind leaders Juventus.
Young's move will continue a recent flow of players from Old Trafford to San Siro, with Alexis Sanchez signing for Inter on loan last August after fellow forward Romelu Lukaku completed a permanent switch.
What makes the move unusual is that Young is unlikely to get many minutes in Inter either as he will probably be a rotational option there too but at least he's the captain of the Red Devils here.