Yes this is possible, following the precedent on opportunity attacks from the Sage Advice Compendium
First I will point out the similarity to the wording of the "Opportunity Attacks" section:
To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature.
Compare this with the Whirlwind Attack feature:
You can use your action to make a melee attack against any number of creatures within 5 feet of you, with a separate attack roll for each target.
These use near-identical wording and thus we can say that rulings the Sage Advice Compendium has made regarding opportunity attacks also apply to Whirlwind Attack.
With that concluded, the Sage Advice Compendium has said the following:
Q. What actions can monsters use to make opportunity attacks? Are Multiattack and breath weapon actions allowed?
A. A monster follows the normal opportunity attack rules (PH, 195), which specify that an attack of opportunity is one melee attack. That means a monster must choose a single melee attack to make, either an attack in its stat block or a generic attack, like an unarmed strike [...]
From the above quote we see that things such as the Bulette's bite attack can be used with opportunity attacks and thus they can also be used with Whirlwind Attack.
A GM has no obligation to agree with the Sage Advice Compendium
The SAC is a collection of official rulings, or ways to interpret rules and these are, of course, able to be contested or changed by any GM. In this case in particular, the SAC ruling doesn't state anything to support its own stance.
Similarly, Whirlwind Attack is a rather exceptional feature (also according only to the SAC) in that it is a single attack despite involving multiple attack rolls. So even if a GM agrees with the SAC's ruling on opportunity attacks, they do not have to apply that same ruling to Whirlwind Attack or other similar cases (we don't even know why it applies to opportunity attacks so technically we can't even extrapolate it to any other cases at all without making some assumptions).
Basically, while the SAC does seem to make this just work, a GM has plenty of room to rule otherwise coming from quite a few differently angles.
