In the leaked recording, Mr. Turnbull joked about the leaders’ first face-to-face meeting last month in New York, which had been preceded by much analysis about its potential awkwardness in light of the January call.

“It was beautiful,” Mr. Turnbull said. “It was the most beautiful putting-me-at-ease ever.”

Many Australian commentators thought Mr. Turnbull had come off as weak during that meeting, and there was speculation Thursday that the recording was leaked deliberately to make him look less subservient to the president.

Mr. Turnbull appeared to take the leak in stride, noting the self-deprecating aspect of his speech.

“It’s a breach of protocol, it’s a breach of faith and all those things, but you know it’s lighthearted, it’s affectionate, good-natured, and the butt of my jokes was myself,” Mr. Turnbull told a radio station.

But he also suggested that he would not be making fun of other world leaders next time.

“I guess what that means is that next year at the Mid Winter Ball, I will read selected passages from Budget Paper No. 2,” Mr. Turnbull said.