I want to know whether there is any interaction between masses due to gravity. To illustrate my point suppose two masses are in space. They will get attracted to each other. But is this interaction due to exchange of gravitons or there is no interaction between them but a natural movement due to curvature of space time?
The word interaction does not have a precise meaning in physics, but to the extent that it does have a meaning it implies an exchange of momentum. That is, if two bodies interact then they exchange momentum with each other. This implies there is an interaction force since change of momentum (also called impulse) is the integral of force with respect to time.
In relativity we should really use four momentum since this is a covariant quantity, but whether you use four momentum or three momentum it is obvious that two massive bodies do exchange momentum with each other and therefore they do interact.
But the question of how momentum is exchanged is one of those what really happens questions that have no answer in physics. Physics is the construction of mathematical models to predict physical behaviour. Newton used the gravitational field as his mathematical model, and quantum field theorists use virtual particles as their mathematical model. Both are perfectly good models within their region of applicability, but in neither case is the model intended to be a definitive statement of what is really happening. (We could at this point make the usual warning that virtual particles do not actually exist.)
So your question has no answer. The gravitational field and exchanges of virtual gravitons are both valid descriptions of the gravitational interaction. However we should note that gravitons are so weakly interacting that we are unlikely ever to detect a (real) graviton. We expect quantum gravity effects to become detectable only at energy densities far beyond what we can achieve in the laboratory.