Mr. Trump first accused Mr. Obama a month ago of tapping his phones at Trump Tower during the campaign last year. He has refused to back down, even though Mr. Obama and his top aides have adamantly denied it. The F.B.I. director and the former director of national intelligence have said the phone tapping charge is not true, and congressional leaders of both parties have said they have seen no evidence of it.

In an interview broadcast on BBC on Monday evening, John O. Brennan, the C.I.A. director under Mr. Obama, chided Mr. Trump for making an unsubstantiated allegation against the former president. Mr. Trump, he said, has “a solemn obligation” to provide information “that is accurate, that is measured and that is not just a spontaneous or impulsive number of words.”

While other officials have said there is no convincing evidence so far of collusion between Mr. Trump’s campaign and Russian officials who meddled in last year’s election, Mr. Brennan said that “it would be premature at this time to make any determination, or rule anything out.” At the same time, he agreed with Mr. Trump about the seriousness of leaks to the news media in recent weeks. “These leaks are appalling,” he said. “They need to stop.”

In trying to combat what Mr. Trump’s aides see as a concerted campaign of leaks to undermine his legitimacy, the White House last month provided intelligence to Representative Devin Nunes, the Republican chairman of the House Intelligence Committee, showing that the president or his associates may have been “incidentally” swept up in foreign surveillance by American spy agencies last year. Since Mr. Nunes made that public, Mr. Trump’s team has focused on whether Mr. Obama’s White House improperly used that information.

Republicans pointed to the reports about Ms. Rice on Monday. “Smoking gun found!” Senator Rand Paul, Republican of Kentucky, wrote on Twitter. “Obama pal and noted dissembler Susan Rice said to have been spying on Trump campaign.”

Intelligence officials are supposed to guard the privacy of Americans caught up in routine eavesdropping of foreign officials. In daily intelligence reports to officials like Ms. Rice, they typically refer to Americans who came up in recorded conversations as U.S. Person One or U.S. Person Two. But high-ranking officials, as Ms. Rice was, can ask intelligence briefers to provide names to better understand the meaning of the report.