One has to keep clear in mind that the standard model, which has the SU(3)xSU(2)xU(1) structure is called standard because it describes extremely well an enormous number of particle data and is successful in its predictions for new experiments. As Luc J. Bourhis explores in his answer this does not stop theorists from exploring beyond the standard model theories.
The standard model is a quantum field theoretical model, which means it has precise predictions for evaluating Feynman diagrams which will be used for fitting or predicting experimental data. In its mathematical structure there is no simple vertex, i.e. lowest order, ( higher order loops can give an interaction) between the electroweak bosons (W, Z, γ) of the table, and the gluon .
Thus by construction of the mathematics of the model the gluon does not "see" the Higgs field. As the very existence of the concept of a gluon depends on the mathematics of the standard model, our "belief" in the standard model means massless gluons. Other theories exist beyond the standard model , which may give a coupling, ( example ) but they are beyond the standard model. Be sure that experiments will be testing for any discrepancy with the standard model that could be due to a massive gluon.
Question in comment:
Could you please elaborate how you jumped from the weak bosons to the Higgs field
It has to do with the fact that Feynman diagrams are formulated with specific rules on the fields of all the elementary particles in the table, including the Higgs field. So in a simple diagram of e-e- scattering,
the electron creation/annihilation operators operate sequentially on the electron field, when a created electron interacts with the photon field and photon creation operators create a virtual photon, which interacts with the electron field and creates an outgoing electron.
Thus in field theory the vertex coupling constant should exist with the field so that the interaction can happen and, in this case, a photon can be generated.
Because of the zero mass there is no coupling constant for the vertex gluon-higgs field, to generate a virtual higgs and continue from there. Only higher quark loops can act , i.e. glue to q q_bar, virtual higgs loop Higgs meson from gluon fusion.
Edit after edit of OP:
This is ok for the Higgs boson but not for the Higgs field
The Higgs has a weak charge and therefore interacts with the W and Z bosons thus giving them mass. The Higgs does not have the electric or color charge and therefore does not interact with the photon or gluons thus leaving them massless.
The mass the elementary particles of the SM table acquire comes from the interaction with the Higgs FIELD, not the Higgs Boson. The Higgs boson is just another massive particle in the elementary particles table:
These are the only masses the Higgs Field generates. including the Higgs boson itself. All other masses come from special relativity relations, the invariant mass of the added fourvectors of composite particles.
The standard model has a Lagrangian describing the SU(3)xSU(2)xU(1) experimentaly observed symmetries in particle data and any comparison depends on feynman diagram calculations within this model, where there are strict rules for the exchange vertices.
Each elementary particle in the table defines a field in four dimensional space, and the particle is considered and excitation on this field. The electron is an excitation of the electron field, the Higgs boson is the exitation of the Higgs field.
The elementary particles in the table do not acquire mass by exchanging anything in the form of feynman diagrams. The mass is acquired once, at the time of symmetry breaking of the electroweak interactions, where the three couplings approach each other :
when the couplings of the weak and electromagnetic change due to the Higgs field. One has to study mathematically this to be convinced, but the fact is , that the standard model as now known describes practically all particle physics data and is very predictive of new ones, as the LHC experience shows.
The gluons are massless by construction the way the photon is, and the way the Z and W were massless before symmetry breaking..
Once again, it is the Higgs Field that gives the masses to the elementary particles , not the boson.