"Apparently I was at a big fat Greek wedding...wish I'd been there! @kylieminogue #rumoursrumoursrumours," he wrote.
Their denial came after Joshua, 28, and Kylie, 48, were believed to have tied the knot in secret last month. According to reports, Kylie's birthday celebrations on the Greek island of Sifnos doubled up as her wedding.
Kylie Minogue and Joshua Sasse were reported to have wed on the Greek island of Sifnos
"It was the best-kept secret in the world," an insider told Australia's Woman's Day, adding that guests from all over the world jetted in for the event.
"Kylie flew them all in first class for what was meant to be an amazing birthday party."
However the Locomotion singer didn't make any reference to her so-called wedding and instead wrote on Twitter: "Oh thank you, THANK YOU for your lovely birthday messages!"
The couple announced their engagement in February
Joshua popped the question earlier this year and the couple's engagement was announced in style in the Daily Telegraph.
The February notice read: "Mr J.S. von Sasse and Miss K.A. Minogue. The engagement is announced between Joshua, of the late Dominic Sasse and of Mary Heale (neé Macauley) of Herefordshire and Kylie, eldest daughter of Ronald and Carol Minogue of Melbourne, Australia."
It is believed that the pair chose to make the announcement through the newspaper because of Joshua's family connections with The Telegraph. His great-grandfather William Berry, 1st Viscount Camrose, bought the newspaper in 1927. He also owned The Sunday Times, and several other news outlets.
Apparently I was at a big fat Greek wedding...wish I'd been there! 📰@kylieminogue #rumoursrumoursrumours
— Joshua Sasse (@joshuasasse) June 6, 2016
Kylie had prompted speculation when she was spotted wearing a large diamond engagement ring at the NME Awards just days before.
Kylie and Joshua met while filming television show Galavant last year. Addressing their 20-year age gap, Kylie said: "I think the general consensus is, we kind of look right together. So even though on paper it might not look right, it just is right."
She added to radio show Nova 96.9's Fitzy and Wippa: "He is an absolutely brilliant, brilliant guy. And even if I try and act cool and natural, it's written all over my face that I am very happy."