"Neither the White House nor the president will comment further until such oversight is conducted," he added.
Congressional investigators are probing suspected Russian efforts to undermine the 2016 election as well as any contacts between Russian officials and the Trump campaign.
Early Saturday, Trump - apparently in response to reports by a conservative radio host and on the conservative website Breitbart - accused Obama of wiretapping Trump Tower and engaging in a "Nixon/Watergate" effort to undermine his presidency.
But a senior U.S. official said Saturday that there was no such wiretap. A spokesman for Obama also said that the former president never authorized a wiretap of Trump or any other American citizen.
Speaking on NBC News on Sunday morning, former director of national intelligence James Clapper, who served in that post in the Obama administration, flatly denied that a wiretap was authorized against Trump or his campaign during his tenure.
"There was no such wiretap activity mounted against the president-elect at the time as a candidate or against his campaign," Clapper said on "Meet the Press."
He added that he would "absolutely" have been informed if the FBI had sought or received a warrant to wiretap Trump or his campaign.
"I can deny it," Clapper continued.
The White House's escalation of Trump's claims were kept at arm's-length by congressional Republicans appearing on Sunday morning news broadcasts.
When asked about Trump's allegations, Senate Intelligence Committee member Tom Cotton, R-Ark., declined to comment on the president's tweets but said he has "seen no evidence of the allegations."
"Whether that's a FISA court application or denial of that application or a re-submission of that application, that doesn't mean that none of these things happened. It just means we haven't seen that yet," Cotton added, speaking on Fox News Sunday.
Sen. Marco Rubio, R-Fla., said he is not aware of any evidence to back up the president's claim.
"I have no insight into exactly what he's referring to," Rubio said on "Meet the Press." "The president put that out there, and now the White House will have to answer for exactly what he was referring to."
Obama's allies were more blunt, denying flatly that the former president had ordered a wiretap of Trump's campaign.
"This may come as a surprise to the current occupant of the Oval Office, but the president of the United States does not have the authority to unilaterally order the wiretapping of American citizens," said former White House press secretary Josh Earnest.
Earnest accused Trump of attempting to distract from the controversy involving contacts between his campaign aides, including now-Attorney General Jeff Sessions, and Russian officials.
"We know exactly why president Trump tweeted what he tweeted," Earnest added.
"Because there is one page in the Trump White House crisis management playbook and that is simply to tweet or say something outrageous to distract from a scandal. And the bigger the scandal, the more outrageous the tweet."
Appearing on ABC's "This Week," White House deputy press secretary Sarah Huckabee Sanders repeatedly said that the president's allegation was worth looking into.
"He's asking that we get down to the bottom of this, let's get the truth here, let's find out," Huckabee Sanders said. "I think the bigger story isn't who reported it but: Is it true? And I think the American people have a right to know if this happened, because if it did, again, this is the largest abuse of power that, I think, we have ever seen."
Asked whether Trump truly believes Obama wiretapped him, Huckabee Sanders deflected.
"I would say that his tweet speaks for itself there," she said.
On the broader question of apparent Russian interference in the 2016 election, Clapper urged congressional investigators to attempt to settle the issue, which he said has become a "distraction" in the political sphere.
The intelligence community found no evidence of collusion between the Trump campaign and the Russian government - at least until the end of the Obama administration, the nation's former top spy said Sunday.
"We had no evidence of such collusion," Clapper said on "Meet the Press."
He added a caveat: "This could have unfolded or become available in the time since I left government."
The revelation that, at least until Jan. 20, the intelligence community had not uncovered evidence of any collusion sheds new light on a central question fuelling the wide-ranging federal probe into Russian interference in the 2016 election.
