The 32-year-old sustained the problem during Swansea’s Championship win over Hull City on September 17 and has not played a competitive match since.
Allen has missed each of Swansea’s last 11 second-tier matches, but he was unsurprisingly selected in Page’s squad for the global tournament in Qatar.
Doubts lingered over the midfielder’s chances of being ready for the opening matchday, though, and Page has now conceded that Allen will not be available against the USA on Monday.
“We said from the start, we’re going to give every possible chance that we can to get him to play that first game,” Page told reporters on Sunday. “We could have pushed him, but if he breaks down then he’s definitely out of the tournament. Risk over reward.”
Barring the absent Allen, Wales are expected to be working with a fully-fit squad for Monday’s game, with Gareth Bale declaring himself ready for the Group B opener at the Ahmad bin Ali Stadium.
Page’s side will then take on Iran on November 25 before a battle of the Brits with England on November 29, and Allen will hope to return for one or both of those fixtures.