"advance a notion" is correct and usable in written English.
You could use this phrase when you want to express an idea, opinion, or point of view that you would like to put forward. For example, "I would like to advance the notion that technology can be used to promote social change."
This paper advances a notion of incivility as a precursor to a continuum of antisocial acts culminating in bullying behaviour.
Wilson advanced a notion that is as significant for moral theory as the notion of norms dictated by universal, evolutionarily acquired preferences and indeed ultimately more useful, namely ambivalence.
It sounds almost too ludicrous to believe anyone would advance such a notion, yet it's all over the intertubes today.
Over the course of his directing career, Mr. Eastwood has quietly advanced a revisionist notion of masculinity, a critique from within of the brutal individualism that made him famous as an actor.
Piccinini and Scarantino (2010) advance a highly general notion of computation which they dub generic computation with that consequence.
The White House is complicit in the current mess for advancing an absolutist notion that self-policed markets self-correct, and therefore require little or no regulation.
He is advancing an interesting notion -- that "Congress" is a body that could, if they wanted to, "pass laws" that affected the budget.
There's a notion, advanced by hawkish types in the US, that Russia is intervening in Syria because Obama left a leadership vacuum in Syria.
"It beggars belief that such a major policy should be advanced with so flimsy a notion of how it will be funded.
A prominent exception was the "nudge," a notion advanced by the legal scholar Cass R. Sunstein, now at Harvard Law School, and the University of Chicago behavioral economist Richard Thaler, in their 2008 best-seller "Nudge: Improving Decisions About Health, Wealth, and Happiness".
