The date of the match will remain the same with Pep Guardiola's side set to meet the Bundesliga club on Feb. 24.
"UEFA would like to thank Borussia VfL 1900 Monchengladbach and Manchester City FC for their close cooperation and support, as well as the Hungarian Football Federation for their assistance and agreeing to host the match in question," the statement read.
The announcement comes after Liverpool's Champions League tie against RB Leipzig next week was moved on Sunday from Germany to neutral Budapest. German authorities declined to give Liverpool an exemption from restrictions limiting travel from England where an aggressive variant of COVID-19 is spreading.
United are due to play the first leg of their last-32 tie in Spain on Feb. 18 but, as things stand, the Spanish government have not yet granted an exemption to allow elite sports teams to travel to the country without having to quarantine on arrival in line with COVID-19 protocols.
Sources have told ESPN that Sociedad are awaiting a response from the Spanish government and are looking at alternative venues to play the match.
Talks between Spanish clubs, the Spanish FA and the Spanish government are ongoing.
Chelsea's Champions League trip to Atletico Madrid is also affected while sources have told ESPN that UEFA preferred stadiums to be used that have a history of hosting European games. Sources added that Bucharest is the first option, with Udine and Verona seen as alternatives.
One of the venues being considered for United's game is Juventus' Allianz Arena in Turin. Any move would need approval from UEFA and the Italian FA and quarantine exemptions for both United and Sociedad would need to be granted by the Italian government.
Sources have told ESPN that United expect to be informed of a decision by Tuesday morning.
UEFA also moved a Europa League game to neutral Spain on Monday because travel restrictions in Norway prevent German club Hoffenheim traveling there next week.
Molde are unable to host Hoffenheim in the first leg of the round of 32, which will be played at Villarreal's stadium as a neutral venue on Feb. 18.
Villarreal are still in the competition and will host Salzburg one week later. It is the second game UEFA has moved this month due to international border controls aimed at controlling the spread of COVID-19 infections.
Information from ESPN's Rob Dawson, Adriana Garcia and Rodrigo Faez and The Associated Press was included in this report.