We investigate the circumstellar disc fraction as determined from L-band excess observations of the young, massive Arches and Quintuplet clusters residing in the central molecular zone of the Milky Way. The Quintuplet cluster was searched for L-band excess sources for the first time. We find a total of 26 excess sources in the Quintuplet cluster, and 21 sources with L-band excesses in the Arches cluster, of which 13 are new detections. With the aid of proper motion membership samples, the disc fraction of the Quintuplet cluster could be derived for the first time to be 4.0 ± 0.7%. There is no evidence for a radially varying disc fraction in this cluster. In the case of the Arches cluster, a disc fraction of 9.2 ± 1.2% approximately out to the cluster's predicted tidal radius, r< 1.5 pc, is observed. This excess fraction is consistent with our previously found disc fraction in the cluster in the radial range 0.3 ⊙, as derived from J-band photospheric magnitudes. We discuss the unexpected finding of dusty circumstellar discs in these UV intense environments in the context of primordial disc survival and formation scenarios of secondary discs. We consider the possibility that the L-band excess sources in the Arches and Quintuplet clusters could be the high-mass counterparts to T Tauri pre-transitional discs. As such a scenario requires a long pre-transitional disc lifetime in a UV intense environment, we suggest that mass transfer discs in binary systems are a likely formation mechanism for the B-star discs observed in these starburst clusters.