The Scot was relieved of his duties on Tuesday morning just 10-months into a six-year contract, having presided over the club's worst campaign of the Premier League era.
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A club statement read: "Following the departure of David Moyes as manager, Manchester United has announced that Ryan Giggs, the club’s most decorated player, will assume responsibility for the first team until a permanent appointment can be made.
"The club will make no further comment on this process until it is concluded."
Borussia Dortmund boss Jurgen Klopp had been installed as the early favourite to succeed Moyes, but has since ruled out the possibility of leaving the Bundesliga giants this summer.
Atletico Madrid manager Diego Simeone and Real Madrid coach Carlo Ancelotti are both also thought to be in contention, but Netherlands boss Louis van Gaal is the new front-runner for the post.
Van Gaal is also wanted by Tottenham to succeed Tim Sherwood, but Goal reported last week that the Dutchman was taking his time to weigh up his options amid speculation linking him with United.