"advanced the notion" is correct and usable in written English.
You can use it when talking about someone bringing up an idea or suggesting a concept. For example, "The professor advanced the notion that education is the key to success."
Du Bois, who first advanced the notion of an "Encyclopedia Africana" in 1909.
For years, ExxonMobil actively advanced the notion that its products had little or no impact on the Earth's environment.
First, Subaru transformed the humdrum Legacy station wagon into the stylishly rugged Outback, then advanced the notion when it hatched the Forester from its Impreza compact car.
He has talked about his battle with depression, compared acting to therapy and repeatedly advanced the notion that comedy is rooted in suffering.
Girolamo Fracastoro, an Italian scholar, advanced the notion as early as the mid-1500s that contagion is an infection that passes from one thing to another.
Whereas recent data have advanced the notion that dilation can be done effectively with minimal morbidity, issues concerning safety and tolerability remain.
Why no Ailey Ken doll to advance the notion of male dancers?
That was more than a decade ago, when Mr. Hawkins was advancing the notion of a huge Federal job-creating program.
Mark Pincus at Zynga was one of the first to advance the notion that in the future all the people would be gaming all the time.
Even so, some legal reformers in China believe that advancing the notion of constitutional law is critical in establishing the rule of law.
Last year, you wrote a book, "Only the Paranoid Survive!," advancing the notion that a company must always be fearful of its competitors.
