Rules as Intended: No
Jeremy Crawford has clarified:
Q: Is warding bond the source of its generated damage instance? If so what damage type is it? Is it still spell damage? thanks
A: When you take damage via warding bond, you're taking damage from whatever caused damage to the target of warding bond.
So, according to Jeremy's clarification, the damage you receive from warding bond is considered the same source as what damaged the ward. Thus, the only time aura of warding would work on it is if that original damage was caused by a spell.
As always, it is up to the DM to decide how much designer clarifications and intent means at their table.