A few weeks after Senator Marco Rubio joined a bipartisan push for an immigration overhaul in 2013, he arrived alongside Senator Chuck Schumer at the executive dining room of News Corporation’s Manhattan headquarters for dinner.

Their mission was to persuade Rupert Murdoch, the owner of the media empire, and Roger Ailes, the chairman and chief executive of its Fox News division, to keep the network’s on-air personalities from savaging the legislation and give it a fighting chance at survival.

Mr. Murdoch, an advocate of immigration reform, and Mr. Ailes, his top lieutenant and the most powerful man in conservative television, agreed at the Jan. 17, 2013, meeting to give the senators some breathing room.

But the media executives, highly attuned to the intensifying anger in the Republican grass roots, warned that the senators also needed to make their case to Rush Limbaugh, the king of conservative talk radio, who held enormous sway with the party’s largely anti-immigrant base.