Eberle was injured during Saturday's game in Dallas after getting tangled up awkwardly with Stars forward Jamie Benn early in the game.
The 21-year-old leads the Oilers with 17 goals and 43 points in 41 games.
His loss is another tough blow for a team already playing without forward Ryan Nugent-Hopkins (separated shoulder) and defencemen Ryan Whitney (ankle) and Tom Gilbert (knee), among others.
On Monday, the Oilers recalled forwards Magnus Paajarvi and Ryan O'Marra from the American Hockey League while placing Gilbert Brule on re-entry waivers.
He can be brought up as well if he goes unclaimed by another team before Tuesday.