On the level of orbits of satellites the gravitomagnetic field of a rotating gravitating body like the Earth manifests itself in the precession of the orbital plane which is known as the Lense-Thirring effect. Here we discuss the question whether this gravitomagnetic field can also be detected using clocks on orbiting satellites. Two clocks on counterro-tating equatorial circular orbits around the Earth show a difference of 10-7 s which is quite large. However, in general the detectability of this effect depends on the accuracy and stability of the used clocks as well as on the precise knowledge of the satellites orbits. We show that with present technology it is possible in principle to measure this gravitomagnetic clock effect. In particular, we analyze whether this gravitomagnetic clock effect has any impact on the clocks on the Galileo satellites and, thus, on Galileo positioning. Such a measurement would constitute another important Solar System test of Einstein's General Relativity.
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