The 31-year-old moved to Old Trafford from Arsenal in a swap deal for Henrikh Mkhitaryan back in 2018, though struggled to replicate his sparkling form for the Gunners.
He was shipped out on loan to Inter last summer and initially struggled upon his return to Serie A, with his performances hampered by a string of injury issues.
However, his form has improved markedly since the season restarted and Inter boss Antonio Conte was very keen to turn the loan spell into a permanent move.
Confirming the deal on Thursday, United said: ‘Everyone at Manchester United wishes Alexis Sanchez all the best in his future career as he joins Inter Milan on a permanent transfer.’
Inter are believed to have paid no fee for Sanchez, with United eager to get his enormous wages – rumoured to be as much as £500,000-a-week – off their salary bill, and he has signed a contract with his new club until 2023.
In 13 Serie A games since the season restarted, Sanchez has scored three times and laid on nine assists, becoming an integral part of the side.
Sanchez’s total of four league goals for Inter actually betters his entire tally for United in the Premier League, with the Chilean only netting three times in the Premier League in 32 appearances across one-and-a-half seasons.
Speaking about the move on Wednesday after United’s win over LASK, Ole Gunnar Solskjaer said: ‘They’ve liked him, so of course that’s a good move for him. Wish him all the best. He’s a top player, that we want to see play his best football.
‘For whatever reason we didn’t see the best of Alexis, but he’s a top professional and we just wish him all the best.’
United and Sanchez could get an almost immediate reunion in the Europa League, with the Red Devils on course to meet Inter in the final of the competition at the end of August, with former United duo Romelu Lukaku and Ashley Young also in the side.
For more stories like this, check our sport page.