During the Putin call, the Russian leader raised the possibility of talks on a number of issues, including the New START treaty limiting nuclear weapons deployments, according to a report Thursday by Reuters.
The news agency said Trump paused to ask aides what the treaty was, and then denounced it as favoring Russia.
Spicer would not comment on some details of the call. But he challenged the report that Trump did not know what the treaty was, saying the president had merely sought an opinion from an adviser during the conversation, which was conducted through a translator.
"It wasn't like he didn't know what was being said," Spicer said of Trump.
A White House statement on the call at the time did not mention any nuclear discussion.
Spicer said Trump is "very concerned" about the continued leaks, which he said represent breaches of protocol and potential illegality.
The New START treaty set limits on both countries' deployed strategic arms. It does not limit non-deployed warheads.
Trump mentioned the treaty, which he called the "start-up," in all three debates with Democrat Hillary Clinton during the presidential campaign. He charged that Russia had increased its number of warheads and said, erroneously, that the United States was not permitted to do the same for non-deployed weapons.
"Our nuclear program has fallen way behind, and they've gone wild with their nuclear program," Trump said during the Oct. 10 debate. "Not good. Our government shouldn't have allowed that to happen. Russia is new in terms of nuclear. We are old. We're tired. We're exhausted in terms of nuclear. A very bad thing."
Current US planning calls for spending more than $1 trillion over the next 30 years to update the American nuclear arsenal.
Jeffrey Lewis, an arms-control expert at the Middlebury Institute of International Studies at Monterey, Calif., said the phone call was troubling because it showed that Trump has not taken the time to learn anything about nuclear policy since the election. "He knows one thing, which is that Obama signed it, so he's going to rail against it," Lewis said.
But the treaty is not without critics.
"I would agree if he said that the treaty is more advantageous to Russia and kind of a bad deal to the United States," said Michaela Dodge, a senior policy analyst with the conservative Heritage Foundation. Trump's call, she said, could mark this as a good time to reexamine whether New START is still good policy for Washington and reshape the debate.