Textbook terminology has changed over the decades on this point. Here is how Wikipedia explains mass in special relativity:
A so-called massless particle (such as a photon, or a theoretical graviton) moves at the speed of light in every frame of reference. In this case there is no transformation that will bring the particle to rest. The total energy of such particles becomes smaller and smaller in frames which move faster and faster in the same direction. As such, they have no rest mass, because they can never be measured in a frame where they are at rest. This property of having no rest mass is what causes these particles to be termed "massless." However, even massless particles have a relativistic mass, which varies with their observed energy in various frames of reference,John Baez explains:
Photons are traditionally said to be massless. This is a figure of speech that physicists use to describe something about how a photon's particle-like properties are described by the language of special relativity.A figure of speech? Yes, you can say that photons have mass, or do not have mass, as long as you define your terms carefully.
Here is a modern opinion:
For many years it was conventional to enter the discussion of dynamics through derivation of the relativistic mass, that is the mass–velocity relation, and this is probably still the dominant mode in textbooks. More recently, however, it has been increasingly recognized that relativistic mass is a troublesome and dubious concept.Obviously Greene was concerned that he might be embarrassed before his colleagues using some antiquated terminology.
It would have been better to say:
Yes, photons have mass and momentum. Some physicists like to say that the rest mass is zero, but the photon is never at rest anyway.Greene is teaching two new courses on "Einstein's special theory of relativity". It is funny that he always has to say that it is Einstein's theory. Everything he mentions was discovered by someone else, not Einstein.