According to sources, Stephen Brady was on the airport tarmac with his partner of 32 years Peter Stephens, waiting to meet Abbott as he arrived in Paris on Anzac Day (April 25).
The prime minister's travelling party sent an instruction that Stephens should not take part in the greeting but should wait in the car, the Sydney Morning Herald reported.
It is understood that no explanation was given.
Word of the incident, described privately by Canberra officials as "bizarre" and "extraordinary", spread quickly through diplomatic circles.
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A spokesman for Abbott did not deny that it had occurred but said: "The prime minister was very happy to be met by Ambassador Brady and his partner when he arrived in Paris last month."
Brady later offered his resignation, though, it is understood that it has been rejected. The ambassador declined to comment, the report said.
Brady and Stephens were the world's first openly gay ambassadorial couple to be formally recognised when the former took up the post of ambassador to Denmark in 1999.
Abbott hosted a farewell dinner for Brady and Stephens when the couple left Canberra to take up the Paris job. The prime minister also invited them to a staff dinner in Paris on April 26.