Two masses m1 and m2 start moving towards each other due to their mutual force of attraction. what will be the change in their respective centre of mass position?

There will be no change centre of mass position because the position of centre of mass of a system of particles does not depend upon the forces on particles.
Center of mass depends on the mass of the body. 

For the particles of different masses the center of mass will lie closer to the heavier body because center of mass is a point about which the whole mass of the system is balanced.  

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the centre of mass remain unchanged
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