In the summer of 2016, government workers in Illinois received a mailing that offered them tips on how to leave their union. By paying a so-called fair-share fee instead of standard union dues, the mailing said, they would no longer be bound by union rules and could not be punished for refusing to strike.

“To put it simply,” the document concluded, “becoming a fair-share payer means you will have more freedom.”

The mailing, sent by a group called the Illinois Policy Institute, may have seemed like disinterested advice. In fact, it was one prong of a broader campaign against public-sector unions, backed by some of the biggest donors on the right. It is an effort that will reach its apex on Monday, when the Supreme Court hears a case that could cripple public-sector unions by allowing the workers they represent to avoid paying fees.

One of the institute’s largest donors is a foundation bankrolled by Richard Uihlein, an Illinois industrialist who has spent millions backing Republican candidates in recent years, including Gov. Scott Walker of Wisconsin, Senator Ted Cruz of Texas and Gov. Bruce Rauner of Illinois.