The Laser Interferometer Space Antenna (LISA) is a joint ESA/NASA mission proposed to observe gravitational waves. One important noise source in the LISA phase measurement will be on-board reference oscillators. An inter-spacecraft clock tone transfer chain will be necessary to remove this non-negligible phase noise in post processing. One of the primary components of this chain are electro-optic modulators (EOMs). At modulation frequencies of 2 GHz, we characterise the excess phase noise of a fibre-coupled integrated EOM in the LISA measurement band (0.1 mHz to 1 Hz). The upper phase noise limit was found to be almost an order of magnitude better than required by the LISA mission. In addition, the EOM's phase dependence on temperature and optical power was determined. The measured coefficients are within a few milliradians per kelvin and per watt respectively and thereby negligible with the expected on-board temperature and laser power stability.