grease somebody's palm
grease (one's) palm
To bribe one discreetly, as by slipping money into their palm. A: "How are we going to get a table without a reservation?" B: "Don't worry, I greased the hostess' palm, and she promised us a table in no time." I greased my brother's palm to stop him from telling Mom that I had a party while she was away.
grease somebody’s ˈpalm(old-fashioned, informal) give somebody money in order to persuade them to do something dishonest: Luckily, Mick was able to grease a few palms, thus helping his brother to escape.
This phrase refers to the fact that you put grease (= a thick substance like oil) on a machine to make it run smoothly.