It does happen all kinds of ways. Generally, though, the producers send scripts to (bankable) actors’ agents, rather than the actors approaching the producers, simply because the script usually comes before the buzz. In other words, in most cases, Tommy Lee Jones won’t know there’s anything to express an interest in until fairly late in the development process. Previous posters have mentioned examples that didn’t work that way, and there are lots more, often to do with famous/respected directors or writers or occasionally production companies.
That’s just how it works for bankable actors, though. Anyone whose Q score isn’t high enough to impress an investor, for whatever reason, may be asked to audition.
And then there’s a fair number of actors who aren’t famous but don’t have to audition much, simply because they’ve worked enough and directors or producers know them personally and think they have the talent and work ethic. I think this happens more often in TV than in movies. Auditioning is kind of a pain in the ass for everyone, so if the part is small or if the director already has someone in mind, everyone is motivated to skip it.
I have an acquaintance who used to get cast in all kinds of medium-to-large “prosthetic forehead” roles in the Star Trek franchise; he’s a great guy, a versatile actor with a great sense of humor, and even the makeup people liked having him around, so he was recycled a lot. Similarly, several other union actors I met in the 90s showed up in small roles in every sitcom of the era a few times, not because they were the only talented actors in LA, but because the assistant producer knew they could be counted on to learn their lines on time every time and be pleasant on long shooting days, etc.