WASHINGTON (AP) — Under fire for peddling a debunked conspiracy theory, President Donald Trump's personal lawyer Rudy Giuliani said Sunday he would cooperate with the House's impeachment inquiry only if Trump gave him permission.

"If he decides that he wants me to testify, of course I'll testify," Giuliani said during a series of television appearances where impeachment dominated the discussions.

But it's not clear whether Rep. Adam Schiff, whose House Intelligence Committee is taking the lead on the impeachment investigation, wants to hear sworn testimony from Giuliani.

Central to the Democratic-led inquiry is Giuliani's effort to have Ukraine conduct a corruption probe into 2020 presidential candidate Joe Biden and his son Hunter's dealings with a Ukrainian energy company. Trump echoed that request in a July call with Ukraine's president that has now led to the impeachment drive examining whether Trump linked U.S. aid to Ukraine in exchange for that probe. There has been no evidence of wrongdoing by either Biden.

For now, Schiff, D-Calif., is working to strike a deal with the anonymous whistleblower whose complaint forms the heart of the proceedings against the 45th president.

For Trump, the head-spinning developments pose a threat like none he's encountered before, even from the special counsel Robert Mueller report over Russia's meddling in the 2016 elections. The release last week of a rough transcript of Trump's July 25 call with Ukrainian President Volodymyr Zelenskiy and the whistleblower's complaint have put Trump's own words and actions under heightened scrutiny. Democrats are waiting to see how the White House responds to congressional demands for testimony and documents.

"If they're going to obstruct, then they're going to increase the likelihood that Congress may feel it necessary to move forward with an article on obstruction," said Schiff, the committee chairman. He said the committee was expecting an agreement for the whistleblower's testimony "very soon."

While Trump was at his club in Sterling, Virginia, his former homeland security adviser suggested that Giuliani would be doing the president a disservice by espousing the false story that Ukraine, and not Russia, meddled in the 2016 U.S. elections.

"I am deeply frustrated with what he and the legal team is doing and repeating that debunked theory to the president. It sticks in his mind when he hears it over and over again," said Tom Bossert, who also was an adviser to President George W. Bush. "That conspiracy theory has got to go, they have to stop with that, it cannot continue to be repeated."

During his TV appearances, Giuliani not only repeated the allegations but also brandished what he said were affidavits that support them and claimed that Trump "was framed by the Democrats."

Trump, in his July 25 conversation with Zelenskiy, made a brief and cryptic reference to CrowdStrike, a security firm hired by the Democratic National Committee. The DNC's network had been hacked and emails were stolen that were subsequently published by WikiLeaks.

Crowdstrike detected, stopped and analyzed the hack five months before the 2016 election and determined that Russian agents were responsible. Its findings were confirmed by FBI investigators. But conspiracy theorists dispute that and claim the hack is evidence that Trump is being persecuted by "the deep state."

"I would like to find out what happened with this whole situation with Ukraine, they say CrowdStrike ... The server, they say Ukraine has it," according to the rough transcript of Trump's call with Ukraine's leader.

The theory espouses that the hack was a setup designed to cast blame on Russia.

Giuliani acknowledged that Ukraine was not to blame for the DNC hack, but that the country peddled misinformation during the campaign.

Bossert, however, said he believes that "this president has not gotten his pound of flesh yet from past grievances on the 2016 investigation," he said of Mueller's investigation. "If he continues to focus on that white whale, it's going to bring him down."

Giuliani appeared on ABC's "This Week" and CBS' "Face the Nation," while Schiff was interview on ABC and NBC's "Meet the Press." Bossert spoke on ABC.