The Armenia international picked up the knock during Thursday's 4-1 UEFA Europa League win over CSKA Moscow and was replaced by Alex Iwobi in the 61st minute.
The Gunners will be pained to lose their midfield maestro, who moved to the Emirates Stadium in January as part of a straight swap with Alexis Sanchez.
The former Borussia Dortmund and Shakhtar Donetsk playmaker scored 11 goals in 41 appearances for United in 2016-17, as well as contributing five assists, but later fell out of favour with Red Devils manager Jose Mourinho.
In his absence, Wenger will look to the likes of Danny Welbeck and Iwobi to provide the ingenuity his side needs.
Mkhitaryan was a particular boon to United's 2016-17 European campaign and scored more than half of his goals (six) for last season during the Red Devils' run to the Europa League title.
It now looks likely Arsenal will miss Mkhitaryan for at least one match, meaning Wenger will be forced to shuffle his deck—not for the first time this season.