Recently, diffuse and extended sources in TeV gamma-rays as well as in X-rays have been detected in the direction of the Galactic globular cluster (GC) Terzan 5. Remarkably, this is among the brightest GCs detected in the GeV regime. The nature of both the TeV and the diffuse X-ray signal from Terzan 5 is not settled yet. These emissions most likely indicate the presence of several non-thermal radiation processes in addition to these giving rise to the GeV signal. The aim of this work is to search for diffuse X-ray emission from the GeV detected GCs M 62, NGC 6388, NGC 6541, M 28, M 80 and NGC 6139 to compare the obtained results with the signal detected from Terzan 5. This study will help to determine whether Terzan 5 stands out amongst other GC or whether a whole population of globular clusters feature similar properties. None of the six GCs show significant diffuse X-ray emission on similar scales as observed from Terzan 5 above the particle and diffuse galactic X-ray background components. The derived upper limits allow to assess the validity of different models that were discussed in the interpretation of the multi-wavelength data of Terzan 5. A scenario based on synchrotron emission from relativistic leptons provided by the millisecond pulsar population can not be securely rejected if a comparable magnetic field strength as in Terzan 5 is assumed for every GC. However, such a scenario seems to be unlikely for NGC 6388 and M 62. An inverse-Compton scenario relying on the presence of a putative GRB remnant with the same properties as the one proposed for Terzan 5 can be ruled out for all of the six GCs. Finally, the assumption that each GC hosts a source with the same luminosity as in Terzan 5 is ruled out for all GCs but NGC 6139. (abridged)
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