Abstract: Suppose that 

 is a real or complex unital Banach *-algebra, 

 is a unital Banach 

-bimodule, and G ∈ 

 is a left separating point of 

. In this paper, we investigate whether the additive mapping 
δ: 

 →

 satisfies the condition 
A,B ∈ 

, 
AB = G ⇒ 
Aδ(
B)+
δ(
A)
B*= 
δ(G) characterize Jordan 
*-derivations. Initially, we prove that if 

 is a real unital 
C*-algebra and 
G = 
I is the unit element in 

, then 
δ (non-necessarily continuous) is a Jordan 
*-derivation. In addition, we prove that if 

 is a real unital 
C*-algebra and 
δ is continuous, then 
δ is a Jordan 
*-derivation. Finally, we show that if 

 is a complex factor von Neumann algebra and 
δ is linear, then 
δ (non-necessarily continuous) is equal to zero.