Conway limped off the pitch during Middlesbrough's 2-1 defeat in the second leg of their play-off semi-final against Southampton.
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The 23-year-old saw a specialist on Friday and was told he needed surgery on his ankle, according to Sky Sports reporter Keith Downie.
Following a bright campaign at Boro, with 13 goals in 50 appearances, Conway was nailed on to make Steve Clarke's Scotland squad.
But injury has derailed his World Cup dream after Scotland qualified for the tournament for the first time since 1998, ending a 28-year wait.
Clarke's side open their World Cup campaign with a clash against Haiti on June 14, and then it's Morocco on June 19 before a mouth-watering showdown with Brazil five days later.