The frequent email exchanges reveal that Mr. Saban, who is Israeli-American and not Hispanic, was at the forefront of the push inside Camp Clinton to pound Republican candidate Donald Trump for his stance against illegal immigration and comments about Mexican criminals — a message echoed in Univision news coverage.

Shortly after Mr. Trump announced his run in June 2015, pledging to build a border wall and accusing Mexico of exporting criminals and rapists, Mr. Saban was on the phone with Mr. Podesta plotting strategy.

“Haim thinks we are under reacting to Trump/Hispanics. Thinks we can get something by standing up for Latinos or attacking R’s for not condemning,” Mr. Podesta said in an email to his campaign colleagues.

“Haim is right — we should be jamming this all the time,” responded Jennifer Palmieri, director of communications for the Clinton campaign.

At Univision, news anchor Jorge Ramos became a chief antagonist of Mr. Trump, challenging his illegal immigration policies as un-American at press conferences and from the anchor desk.

Beyond blurring the lines between news and advocacy, Mr. Saban’s prominent role inside the Clinton campaign demonstrated how wealthy donors can buy a seat at the table.

Indeed, Mr. Saban in May 2015 hosted a dinner in the private wine gallery of the posh kosher steakhouse Reserve Cut in New York City for a who’s who of the Clinton universe.

The guest list included Mr. Podesta, Ms. Palmieri, campaign vice chair Huma Abedin, campaign manager Robby Mook, campaign finance director Dennis Cheng, fundraiser Laura Hartigan, deputy national political director Brynne Craig and rapid response director Adrienne Elrod.

Vanderbilt University political science professor Marc J. Hetherington said it wasn’t surprising to find intimate dealings between politicians and big donors such as Mr. Saban, who has poured more than $6 million into Mrs. Clinton’s campaign, super PACs and allied party committees.

That’s the way democracy in America works, he said, adding that it was still troubling.

“This is something that the American public is concerned about — overly cozy relationships between people who have interests in different policies and policymakers,” said Mr. Hetherington, an expert on media coverage of political campaigns and voter behavior. “The concern would be whether there was some quid pro quo.”

The emails did not indicate pressure form Mr. Saban to change Mrs. Clinton’s positions. He appears to believe that he shared her agenda.