Context. Reports on a transient source about 1.25 degrees south of the Galactic Centre motivated these follow-up observations with the WSRT and the reinvestigation of archival VLA data. The source GCRT J1745-3009 was detected during a 2002 Galactic Centre monitoring programme with the VLA at 92 cm by five powerful 10-min bursts with a 77-min recurrence while apparently lacking any interburst emission. Aims. The WSRT observations were performed and archival VLA data reduced to detect GCRT J1745-3009 again at different epochs and frequencies, to constrain its distance, and to determine its nature. We attempted to extract a more accurate lightcurve from the discovery dataset of GCRT J1745-3009 to rule out some of the models that have been suggested. We also investigated the transient behaviour of a nearby source. Methods. The WSRT data were taken in the "maxi-short" configuration, using 10 s integrations, on 2005 March 24 at 92 cm and on 2005 May 14/15 at 21 cm. Five of the six VLA observations we reduced are the oldest of this field in this band. Results. GCRT J1745-3009 was not redetected. With the WSRT we reached an rms sensitivity of 0.21 mJy beam(-1) at 21 cm and 3.7 mJy beam(-1) at 92 cm. Reanalysis of the discovery observation data resulted in a more accurate and more complete lightcurve. The five bursts appear to have the same shape: a steep rise, a more gradual brightening, and a steep decay. We found variations in burst duration of order similar or equal to 3%. We improved the accuracy of the recurrence period of the bursts by an order of magnitude: 77.012 +/- 0.021 min. We found no evidence of aperiodicity. We derived a very steep spectral index: alpha = -6.5 +/- 3.4. We improved the 5 sigma upper limits for interburst emission and fractional circular polarisation to 31 mJy beam(-1) and 8%, respectively. Any transient behaviour of a nearby source could not be established. Conclusions. Models that predict symmetric bursts can be ruled out, but rotating systems are favoured, because their periodicity is precise. Scattering constraints imply that GCRT J1745-3009 cannot be located far beyond the GC. If this source is an incoherent emitter and not moving at a relativistic velocity, it must be closer than 14 pc.