An initial fee of £1.3m (€1.5m) was agreed on Thursday between the two clubs for the 34-year-old.
Young, who is to be replaced as United club captain by Harry Maguire, is expected to finalise his transfer on Friday.
The former Aston Villa star has spent eight and a half years at United, winning four major trophies, including the 2012-13 Premier League title.
The England international made 41 appearances in all competitions in 2018-19 but has been used sparingly by manager Ole Gunnar Solskjaer this season.
Ole has confirmed that #MUFC defender Ashley Young is on the brink of signing for Inter Milan.— Manchester United (@ManUtd) January 17, 2020
Young has played in 12 league games, three Europa League matches and three times in domestic cup competition.
A full-back for much of the past three seasons, his opportunities for regular first-team football have been limited by the signing of Aaron Wan-Bissaka and the emergence of Brandon Williams as competition for left-back Luke Shaw.
Young will join former United team-mates Romelu Lukaku and Alexis Sanchez under Antonio Conte at Inter, who are two points behind Juventus in the Serie A table and bidding for their first domestic title since 2010.