Or when you do the day racing, watch who you race with.
When you approach the guy and honk at him, you see his name on the top left corner just below yours.
Then go to the three bars on the top of your screen in the garage. This is the club info section. There you will see how many other guys you have to beat at night (or get beaten in purpose) to reach him.
BTW, if I noticed correctly while playing, be sure that your pestige rating is high if you want to race for the car.

I have just done an experiment and figured out how pink slips actually work.
I started a new carreer and didn't race anyone. At all.
All I did was teasing cops. We all know that losing them gives big prestige points. So I got up to over 200 and went for a first night race (Green garage, 20th place, prestige 212).
The first guy in the line offered me his car. Main thing is to invite a guy who has less han 100 prestige points when you have over 200. The second guy had over a hundred, I had 255. He offered me money only. When the same guy loses to you multiple times, his prestige drops. Send him under a hundred and consider his car yours. Watch your prestige as if it gets very high, you will start losing it even though you win the race. The only way to bring it up is to lose cops.