When the atom was first measured it was assumed to be composed of rest mass.

As time and discoveries were made it was found that the mass of an atom was not all rest mass but that objects in the atom move with respect to the reference of the person doing the measuring.

Each time another moving item is found in an atom the rest mass assumption is proved incorrect and a portion of the mass formally considered rest mass must be reassigned as mass due to motion with respect to the observer.

The mass due to motion with respect to an observer approaches 0 as the velocity approaches 0.
The mass due to motion with respect to an observer approaches infinity as the velocity approaches the speed of light.

If a photon has a zero rest mass and therefore is composed entirely of momentum mass.
Then it is possible to have no rest mass whatsoever if all mass particles are ultimately composed of zero rest mass photons.

This is about as clear as I can make it.
If you wish some site references let me know, perhaps they may be able to explain it more clearly.

We do not know for sure if a photon actually does have a small rest mass or not, but we do know that it is too small to measure at the present time, which must be very small indeed.

Duane