FAA AIM Sec 5-4-3 par b.1.(d) & (e):
(d) In the case of aircraft already inbound on the final approach course, approach clearance will be issued prior to the aircraft reaching the final approach fix. When established inbound on the final approach course, radar separation will be maintained and the pilot will be expected to complete the approach utilizing the approach aid designated in the clearance (ILS, MLS, VOR, radio beacons, etc.) as the primary means of navigation. Therefore, once established on the final approach course, pilots must not deviate from it unless a clearance to do so is received from ATC.
(e) After passing the final approach fix on final approach, aircraft are expected to continue inbound on the final approach course and complete the approach or effect the missed approach procedure published for that airport.
and Sec 5-4-5 par a.3.(a):
(a) Straight-in IAPs are identified by the navigational system providing the final approach guidance and the runway to which the approach is aligned (e.g., VOR RWY 13). Circling only approaches are identified by the navigational system providing final approach guidance and a letter (e.g., VOR A). More than one navigational system separated by a slash indicates that more than one type of equipment must be used to execute the final approach (e.g., VOR/DME RWY 31). More than one navigational system separated by the word "or" indicates either type of equipment may be used to execute the final approach (e.g., VOR or GPS RWY 15).
If you are on an ILS and the Glideslope failed, AND the NPA is the localizer-only (LOC) approach to the same runway, AND the LOC profile and minimums are published, AND the current weather is above LOC minimums, AND you have not passed the MDA for the LOC approach, AND you have briefed the contingency, then you could continue with the LOC approach. Similarly, if you have RNAV approach capability/certification AND the RNAV/GPS approach is an overlay of the ILS approach on the same approach plate (e.g., "ILS or RNAV RWY...), you could continue with the RNAV approach. In either of those cases, you would be dealing with an external equipment failure, and would continue with a published, degraded portion of the same approach.
However, if you were cleared for the ILS approach, you could not simply transition to a VOR or NDB approach.
AIM Sec 5-4-7 par d:
d. At times ATC may not specify a particular approach procedure in the clearance, but will state "CLEARED APPROACH." Such clearance indicates that the pilot may execute any one of the authorized IAPs for that airport. This clearance does not constitute approval for the pilot to execute a contact approach or a visual approach.
If you were "cleared for approach" (i.e., cleared for any approach to the runway), you could transition to another approach. I doubt I would attempt to do so unless I was prior to the final approach fix AND had the other NavAid set up already for backup/monitoring purposes.
In any case, I would advise ATC of the equipment (Glideslope) failure and my intentions (continue with LOC). If they don't like that, they can give you other clearance.