Rudy Giuliani said on Sunday he would co-operate with the impeachment inquiry only if given permission by US President Donald Trump. "If he decides that he wants me to testify, of course I'll testify," Mr Giuliani, Mr Trump's personal lawyer, said in TV appearances. But it is not clear whether Adam Schiff, whose House intelligence committee is leading the impeachment investigation, wants to hear sworn testimony from Mr Giuliani. The inquiry is looking into Mr Giuliani's effort to have Ukraine conduct a corruption investigation into 2020 presidential candidate Joe Biden and his son Hunter's dealings with a Ukrainian energy company. Mr Trump echoed that request in a July call to Ukrainian President Volodymyr Zelenskiy, leading to the investigation as to whether Mr Trump linked aid to Ukraine to the help he asked for. There has been no evidence of wrongdoing by Mr Biden or his son. Mr Schiff is trying to strike a deal with the anonymous whistle-blower whose complaint led to the proceedings against the president. The release last week of a rough transcript of Mr Trump's July 25 call with Mr Zelenskiy and the whistle-blower's complaint have put Mr Trump's own words and actions under more 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," Mr Schiff said. He said the committee was expecting an agreement for the whistle-blower's testimony "very soon". The threat to Mr Trump is considered to be even greater than that from special counsel Robert Mueller's report over Russia's meddling in the 2016 elections. Former homeland security adviser Tom Bossert suggested Mr Giuliani would be doing Mr Trump a disservice by supporting the false story that Ukraine, and not Russia, meddled in the 2016 US elections. "I am deeply frustrated with what he and the legal team are doing and repeating that debunked theory to the president," said Mr Bossert, who was also an adviser to George W Bush. "It sticks in his mind when he hears it over and over again. That conspiracy theory has got to go, they have to stop with that. It cannot continue to be repeated." On TV, Mr Giuliani repeated the allegations and waved what he said were affidavits that support them, and claimed that Mr Trump "was framed by the Democrats". Mr Trump, in his conversation with Mr Zelenskiy, made a brief and cryptic reference to CrowdStrike, a security company hired by the Democratic National Committee. The committee's network had been hacked and emails were stolen that were later published by WikiLeaks. CrowdStrike detected, stopped and analysed 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 Mr Trump is being persecuted by "the deep state." "I would like to find out what happened with this whole situation with Ukraine," the rough transcript of Mr Trump's call to the Ukrainian president says. "They say CrowdStrike ... the server, they say Ukraine has it." The theory claims that the hack was a set-up to put blame on Russia. Mr Giuliani said that Ukraine was not to blame for the hack, but that it peddled misinformation during the campaign. But Mr Bossert believed that "this president has not got his pound of flesh yet from past grievances on the 2016 [Mueller] investigation. "If he continues to focus on that white whale, it's going to bring him down." Polls suggest public support is growing for the impeachment inquiry, with a new CBS survey showing 55 per cent of Americans and nine in 10 Democrats approve. A separate ABC News/Ipsos poll showed two thirds of Americans considered Mr Trump's call to Mr Zelenskiy to be a serious issue.