The usually grim-faced media mogul practically swooned in his seat. Moments after Jeb Bush delivered what many in the audience described as an unremarkable talk at a conference in Washington, Rupert Murdoch turned to his seatmate, Valerie Jarrett, the White House adviser, to gush over its content and tone.

Mr. Murdoch was pleased that Mr. Bush, the former governor of Florida, had listed the economic benefits of overhauling the nation’s immigration system, confiding in Ms. Jarrett that Mr. Bush, a likely Republican presidential candidate, had said all the right things on the difficult issue, according to three people with firsthand knowledge of the conversation.

It was the kind of warm embrace, from the powerful and widely courted owner of The Wall Street Journal and Fox News Channel, that Mr. Murdoch denied Mitt Romney during his 2012 bid for the White House — a snub that Mr. Murdoch is already signaling he will repeat if Mr. Romney runs in 2016.

In the delicate and unseen campaign underway for Mr. Murdoch’s affections in the next presidential campaign, this much is clear: Mr. Romney is out of the running, a reality that has pained and angered his allies.