Beyond the immediate disconnect, though, is a deeper strain between Mr. Trump and Ms. Haley, according to administration officials and other insiders. Ms. Haley has been perhaps the most hawkish voice on Russia on a team headed by a president who has emphasized his fervent desire for friendship with President Vladimir V. Putin…

At one point recently, he saw Ms. Haley on television sharply criticizing Russia over its intervention in Ukraine. “Who wrote that for her?” Mr. Trump yelled angrily at the screen, according to people briefed on the moment. “Who wrote that for her?”…

Republicans close to the White House whisper about the prospect of an alliance between Ms. Haley and Vice President Mike Pence, possibly to run as a ticket in 2020.

Aides to both scoff at such suggestions, but the slightest hint of such a pairing would be likely to enrage Mr. Trump, who has made it clear that he plans to run for re-election. The talk was exacerbated in recent days when Mr. Pence named Jon Lerner, Ms. Haley’s deputy, as his new national security adviser, while allowing him to keep his job at the United Nations.