A Melee attack includes the ability to apply Reach. However, if they are Adjacent to you already (part of the ability requirements) you won't need Reach to start with.
For each hostile figure adjacent to you, perform 1 attack with a melee weapon targeting that figure
(Emphasis mine.) To initiate this ability, adjacency is first required. Reach does not grant Adjacent status.
It would only apply in this situation if you somehow where able to be Adjacent to a figure while still being 2 spaces away and not sharing an edge—against the standard rules. I'm not away of any cards or abilities that enables that though. (But I don't know all the cards by memory though.)
RRG pg 4:
A space is adjacent to each other space that shares an edge or corner with the space.
• Spaces on either side of the diagonal intersection of walls and/or blocking terrain are not adjacent to each other
Just in case, if you do apply it for some reason, don't forget Reach still requires LoS.
RRG pg 21:
A figure with [Reach] may perform melee attacks that target figures or objects up to 2 spaces away.
• The attacking figure must have line of sight to the target in order to make this attack
In summary, no. You could apply Reach to a Melee attack but since you won't be able to initiate the attack without being Adjacent to start with, Reach won't need to be applied anyway.
(At least, this is my interpretation. Of course, the rules also doesn't explicitly state you can't do this. And it isn't super clear in the Reach rules the exact way it boosts a melee attack to enable it to not be adjacent anymore. So I totally get your confusion. And I wouldn't blame you if you disagree with me, because I can also see enough ambiguity in the rules to argue the other way. It might just come down to agreeing with the other players on a house rule about this one.)