PARIS — The troubles plaguing Boeing after the yearlong grounding of its 737 Max plane have created an unusual opening for its chief rival, Airbus, to swoop in and grab business. There’s just one hitch: Airbus is in no position to benefit.

The European aerospace giant, which last year took the title from Boeing as the world’s biggest plane maker, has such a large backlog of orders to fill that it cannot immediately produce more of its popular narrow-body jets that airlines view as an alternative to the Max.

“It might look like a paradox, but in the short term, we don’t benefit from the situation with our competitor,” Airbus’s chief executive, Guillaume Faury, said Thursday at a news conference in Toulouse, France, announcing the company’s 2019 annual results.

Airbus has been unable to take advantage of the shortfall at Boeing partly because it can’t build planes fast enough. Production of Airbus’s A320 jets — the main competitor to the 737 Max and the bulk of Airbus’s commercial business — is months behind schedule because of slowdowns at some of its European factories.