Green’s Dictionary of Slang

in possession of a large amount of something, usu. money.

1922		Joyce Ulysses 7: He’s stinking with money and thinks you’re not a gentleman.
1939		P. Di Donato Christ in Concrete 93: The Church? Or the City Hall stinking with thieves?
1950		‘Neville Shute’ Town Like Alice 186: Then when they got the water it was stinking with the minerals and the cattle wouldn’t touch it.
1961		F. Kohner Gidget Goes Hawaiian 56: He is stinking with style!
1975		‘P.B. Yuill’ Hazell and the Three-card Trick (1977) 155: I was supposed to be stinkin’ wiv money wasn’t I?