Mkhitaryan joined United for around £30 million in 2016 after three years at Borussia Dortmund.
The 20-year-old forward has attracted interest from the likes of Manchester United and Arsenal this January, but the Bundesliga club are hopeful that the Jamaica-born player, who has a contract until 2022, will elected to commit his future to them. He also revealed he had discussed his dream of playing for the "biggest club in England" with the Old Trafford legend Sir Alex Ferguson.
"So yes, I'm fulfilling a dream and I hope to give my very best and win many trophies at this club".
He wrote: "There are people (former club players) who have spoken with no knowledge of what happens inside the club and cause damage".
Arsenal allowed Sanchez to join Manchester United on Monday to avoid losing the Chile global on a free transfer when his contract expired at the end of the season.
Arsenal boss Wenger said he was happy to acquire Mkhitaryan in the deal.
City in August previous year came close to signing Sanchez and looked to be the favorite to get him this month, only to end their interest in the forward following a rival bid from United.
Arsene Wenger is in very real danger of leaving the club in a worse position than he found it which is downright incredible after all he did for them.
In a statement issued by United, Sanchez said he had spent "three-and-a-half wonderful years" at Arsenal "and I bring with me very positive memories of that great club and its fans".
Sanchez could make his debut for United when they face Yeovil Town in the FA Cup fourth round on Friday night.
Mkhitaryan is cup-tied in that competition, so could make his debut with Arsenal against Swansea City next week. Only Romelu Lukaku, Sergio Aguero and Harry Kane have scored more.
In an Instagram post, he took a parting shot at ex-Arsenal players who have criticised his attitude.
Mkhitaryan has made 63 appearances for United - 22 this season - scoring 13 goals.
If you compare United's squad to City's, it is rather tame.
"He indicated it is important for the club for me to be here".
Ex-Manchester United and Stoke defender Danny Higginbotham: "He's that match-winner".