IGR J17361-4441 is a newly discovered INTEGRAL hard X-ray transient, located in the globular cluster NGC 6388. We report here the results of the X-ray and radio observations performed with Swift, INTEGRAL, RXTE, and the Australia Telescope Compact Array (ATCA) after the discovery of the source on 2011 August 11. In the X-ray domain, IGR J17361-4441 showed virtually constant flux and spectral parameters up to 18 days from the onset of the outburst. The broad-band (0.5-100 keV) spectrum of the source could be reasonably well described by using an absorbed power-law component with a high energy cut-off (NH ~= 0.8 × 1022 cm-2, Gamma ~= 0.7-1.0, and Ecut ~= 25 keV) and displayed some evidence of a soft component below ~2 keV. No coherent timing features were found in the RXTE data. The ATCA observation did not detect significant radio emission from IGR J17361-4441, and provided the most stringent upper limit (rms 14.1 muJy at 5.5 GHz) to date on the presence of any radio source close to the NGC 6388 center of gravity. The improved position of IGR J17361-4441 in outburst determined from a recent target of opportunity observation with Chandra, together with the X-ray flux and radio upper limits measured in the direction of the source, argue against its association with the putative intermediate-mass black hole residing in the globular cluster and with the general hypothesis that the INTEGRAL source is a black hole candidate. IGR J17361-4441 might be more likely a new X-ray binary hosting an accreting neutron star. The ATCA radio non-detection also permits us to derive an upper limit to the mass of the suspected intermediate massive black hole in NGC 6388 of ≲ 600 M&sun;. This is a factor of 2.5 lower than the limit reported previously.