Yes, a real competing-offer deadline is a reasonable reason to contact recruiting. Keep the message factual: you remain very interested, you have another decision deadline, and you are asking whether any acceleration is possible.
Do not frame it as an ultimatum. The consulting-offer next-steps guide is useful for thinking through how to communicate when timelines collide.
Recruiting may ask for the exact deadline and may or may not be able to change the process. That depends on interviewer availability and where you are in the funnel.
If they cannot accelerate, you still need to decide whether to accept the other offer, request an extension there, or take the risk of waiting. Surface the conflict early rather than the night before the deadline.