Do photons feel gravity of approaching objects only? I have read that photons while travelling near massive objects such as the sun experience gravitational pull which is why we see some stars at different positions than they are when seen towards the sun.
Now if photons felt gravity of objects they approach, speed of propagation of gravitational field can be both lesser than or equal to speed of light, but if photons that are receding from objects also feel it's gravity than certainly speed of propagation of gravity would be more than speed of light !
So the question is do photons feel gravity of object they move away from ? If they do is my assertion correct ? 
 A: It's better to think of the deflection of the photon as an effect of its travel through curved spacetime. You can generally choose to analyze the problem in the rest frame of the massive object.  In that case spacetime is curved in all directions around the object, and so the photon's path is deflected both as it approaches and as it recedes.
If you want to think of the gravitational field as an effect of gravitons travelling at $c$, you have to remember that the massive object doesn't come into existence as the photon passes it: the photon can feel the gravitational force "emitted" by the mass in the past. (I feel a little squicky talking about gravitational attraction as something that is "emitted.")
Now, if the massive object were to begin to oscillate after the photon had passed, the gravitational waves emitted by the oscillating object wouldn't be able to catch up with the photon. But that's a more subtle situation than the one in your question.
A: Any object that has mass just has a gravitational field around it and light (or particles, photons) just get refracted when it enters this field.
