He is a "friend of the family", was invited to the royal wedding, and has reportedly holidayed with Prince Albert and Princess Charlene.
So the rumours may be just that - rumours.
She's a flighty one, that Charlene, though.
I was on a cruise a couple of years ago and we couldn't sail into Monaco as preparations for the wedding of Albert and Charlene were in overdrive and the port was closed.
We headed to Nice and the next day the papers were full of the news that Charlene had done a runner to the airport - just days before her wedding - and had to be persuaded to return by palace officials.
She became known in the tabloids as the runaway bride.
She and her husband had the last laugh, however, when they successfully sued the Sunday Times in London for reporting her alleged escape attempts.
So these latest rumours may again be the work of the malevolent forces in Monaco that wish Albert's family harm, as the Prince has claimed.
All I know is that if they're true, Byron is punching waaaaay above his weight.
If Princess Charlene and Byron are really getting it on, and their respective attractiveness ratio is a universal rule of thumb, I have a bloody good chance of having a dart down the blindside with George Clooney.