Following the precedent of the Sage Advice Compendium, Whirlwind Attack will not allow you to make a melee spell attack
The section on Melee Attacks states:
A melee attack typically uses a handheld weapon such as a sword, a warhammer, or an axe. A typical monster makes a melee attack when it strikes with its claws, horns, teeth, tentacles, or other body part. A few spells also involve making a melee attack.
TheThus we do know that certain spells are indeed considered melee attacks and so it would seem Whirlwind Attack would let us use them.
However, this is also the case if we were to look at the section on Opportunity Attacks which states:
You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature.
So itIt would seem we couldcan use melee spell attacks for opportunity attacks; however, the Sage Advice Compendium has stated the following:
Q. Can you use a melee spell attack to make an opportunity attack?
A. You can’t if the spell attack is created by casting a spell. You can’t if the spell attack is created by casting a spell. When a creature triggers an opportunity attack from you, you can use your reaction to make a melee attack against it. The opportunity attack doesn’t suddenly give you the ability to cast a spell, such as shocking graspshocking grasp.
This is also showcased in this Q/A ("Can I use a spell to make an Opportunity Attack?").
Here we see the precedent set, that even though opportunity attacks allow you to make a melee attack, they cannot be used to make melee spell attacks.
Similarly,Following this logic we can conclude that even though the Whilrwind Attack feature allows us to make a melee attack but, it cannot be used to make a melee spell attack.