Is there any relationship between Gravity and Electromagnetism? We all know that the universe is governed by four Fundamental Forces which are The strong force , The weak force , The electromagnetic force and The gravitational force .
Now, is there any relationship between Electromagnetism and gravity?
 A: On Unification
I presume you're asking  whether just classical gravity & classical     EM can be unified.
They sure can!
Classical General Relativity and Classical Electromagnetism are unified in Kaluza-Klein-Theory, which proves that 5-dimensional general relativity is equivalent to    4-dimensional general relativity plus 4-dimensional maxwell equations.      Rather interesting, isn't it? A byproduct is the scalar "Radion" or "Dilaton" which appears due to the  "55" component of the metric tensor. In other words, the Kaluza-Klein metric tensor equals the GR metric tensor with maxwell stuff on the right and at the bottom; BUT you have an extra field down there.     
$${g_{\mu \nu }} = \left[ {\begin{array}{*{20}{c}}
  {{g_{11}}}&{{g_{12}}}&{{g_{13}}}&{{g_{14}}}&{{g_{15}}} \\ 
  {{g_{21}}}&{{g_{22}}}&{{g_{23}}}&{{g_{24}}}&{{g_{25}}} \\ 
  {{g_{31}}}&{{g_{32}}}&{{g_{33}}}&{{g_{34}}}&{{g_{35}}} \\ 
  {{g_{41}}}&{{g_{42}}}&{{g_{43}}}&{{g_{44}}}&{{g_{45}}} \\ 
  {{g_{51}}}&{{g_{52}}}&{{g_{53}}}&{{g_{54}}}&{{g_{55}}} 
\end{array}} \right]$$     
Imagine 2 imaginary lines now.   
$${g_{\mu \nu }} = \left[ {\begin{array}{*{20}{cccc|c}}
  {{g_{11}}}&{{g_{12}}}&{{g_{13}}}&{{g_{14}}} & {{g_{15}}} \\       
  {{g_{21}}}&{{g_{22}}}&{{g_{23}}}&{{g_{24}}} & {{g_{25}}} \\ 
  {{g_{31}}}&{{g_{32}}}&{{g_{33}}}&{{g_{34}}} & {{g_{35}}} \\ 
  {{g_{41}}}&{{g_{42}}}&{{g_{43}}}&{{g_{44}}} & {{g_{45}}} \\ 
\hline
  {{g_{51}}}&{{g_{52}}}&{{g_{53}}}&{{g_{54}}} & {{g_{55}}} 
\end{array}} \right]$$ 
So the        stuff on the top-left is the GR metric for gravity, and the stuff on the edge ($g_{j5}$ and $g_{5j}$) is for electromagnetism    and you have an additional component on the bottom right. This is the radion/dilaton.           
An extension to kaluza - klein is supergravity, which     also talks about the weak and strong forces, and requires supersymmetry.   
On Geometry
In quantum-electrodynamics, the gauge group for electromagnetism is $U(1)$. 
Now, the key thing here is that Electromagnetism is then  The Curvature of the $U(1)$ bundle.                    
This is not the only geometric connection between           General Relativity and Quantum Field Theory. In the same context,         the covariant derivatives is general relativity     are such that $\nabla_\mu-\partial_\mu$ sort-of measures the gravity, in a certain way, while this is also true in QFT, where to some constants, $\nabla_\mu-\partial_\mu=ig_sA_\mu$.           
It is to be noted that both are in similiar context.   
