The 50-year-old pop star has enjoyed a hugely successful solo career since leaving Take That in 1995 but is said to have left organisers "absolutely gutted" after rejecting a big money offer for a stint at Voltaire.
"Robbie is admired by the Voltaire team professionally and personally," a source told The Sun newspaper.
"They love his music and stagecraft and thought that his super stage personality and charisma seemed to be the perfect fit for them.
"But when they packaged up the offer, Rob turned them down.
"They were absolutely gutted as the tickets would have sold out immediately but Robbie seems to be wanting to take a more careful approach to what he does next."
If Williams had accepted the offer, he would have taken the slot recently played by Kylie Minogue, who he collaborated with on the 2000 hit Kids from his album Sing When You're Winning.
She is thought to have earned $A1.5 million per show for her More Than Just A Residency concert, which came after she enjoyed renewed success in the US with her hit single Padam Padam and subsequent album Tension.
Other stars to have graced the Sin City strip include Britney Spears, who played her Piece of Me show in the city from 2013 until 2017 for a reported $US300,000 ($A438,000) per night, while Adele is currently thought to be raking in an estimated $US1.5 million ($A2.2 million) per show for her Weekends With Adele, from which she is currently on a short break.
Angels singer Williams recently admitted he had never really had the chance to crack the US, and had sought anonymity there after enjoying such a public career in his home country, although he is hopeful that success across the pond could come one day.
"I am completely anonymous in America, and I didn't plan to change it until now," he said.
"Now I'm kind of desperate for that to change.
"I moved so I could be Bruce Wayne in America and Batman everywhere else.
"It worked for me because there were periods in my life which were pretty risque.
"I needed a place to retreat and not be known, and I got that.
"But now I'm old and wise enough to want the experience of incredible success again - and maybe that would be in America."
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