May, who has been in sparkling form for his new club Leicester this season with 10 tries in nine games, suffered the injury at England's training camp in Portugal.
"Jonny is doing everything he can to get back fit as quickly as possible. He has been playing really well, but what it means is there are opportunities for other players," England assistant coach Steve Borthwick said.
"We are assessing the situation and getting an understanding of where all the players are at and then I'm sure (coach) Eddie (Jones) will make a decision."
Jack Nowell and Elliot Daly are also injured, leaving Anthony Watson, Semesa Rokoduguni and Denny Solomona as Jones's wing options for the Argentina game on Nov. 11.
Lock Joe Launchbury has a knee problem and is also doubtful for the match against the Pumas.
England also play Australia and Samoa at Twickenham this month.
(Reporting by Ed Osmond; Editing by Toby Davis)
