Dammit. what is that word?
I’m going to try to quietly campaign to be selected to go on a trip. Back room politicking? Negotiating? No, that isn’t the word…
Scheming
Conniving
Dropping hints
Social engineering
Plotting
I don’t think those are it but that might help.
using subtle hints
using gentle influence
suggesting indirectly
prodding
probing the thoughts of someone
(none of 'em are quite right, but that’s all I got .)
pushing
insinuating
manoeuvring
propagandising
advocating
championing
<marlon brando>
Make him an offer he can’t refuse.
</marlon brando>
Working behind the scenes?
lobbying?
That has my vote!
I believe maneuver is the word you’re looking for.
The way I picture it is quietly talking to one or two people at a time gently persuading them of the merits of your idea until most people are agreeing with you (even if it’s just “in theory”) and only after all that do you openly challenge the issue or ask the question in a public forum.
Whisper campaign.
Moving in silently, downwind and out of sight, you gotta strike when the moment is right without thinking.
After a while, can I work on points for style?
I agree. “Manoeuvring” has a connotation that it isn’t done in the open. “Lobbying” on the other hand, can be done either openly or behind the scenes, and thus doesn’t have the sense of being done behind closed doors that it sound like the OP is looking for.
IME, ‘Blackmailing’ has been a winner.
I <3 you.
angling
The club tie, and the firm handshake, a certain look in the eye and an easy smile.
It’s great to be trusted by the people that I post with! 
Machinations?
Finagle, finagling