Two Soviet-born Americans who form a key link between Donald Trump and Ukraine “weren’t James Bond”, the president’s lawyer, Rudy Giuliani, said on Saturday, insisting the men “didn’t have personal communications with the president” during a time period under intense scrutiny in the impeachment inquiry.

Giuliani also repeated a remark made to the Guardian earlier this month, that he is not worried about Trump throwing him under the bus as an impeachment vote and Senate trial loom, because he has “insurance”.

Some observers have treated such remarks as potential veiled threats to Trump, to flip and tell House investigators everything he knows. Others, including Giuliani’s own lawyer, have insisted the former New York mayor is joking.

Lev Parnas and Igor Fruman played key roles earlier this year as Giuliani tried – at the president’s direction – to get Ukraine to investigate both Democratic presidential contender Joe Biden and his son and a discredited conspiracy theory about supposed Ukrainian interference in the 2016 US election.

Nearly $400m in military aid was held up with a White House meeting for the Ukrainian president, Volodymyr Zelenskiy, also dangled as bait. That quid pro quo, confirmed in public testimony by Trump appointee Gordon Sondland, lies at the heart of what Democrats say is an abuse of power that merits Trump’s impeachment and removal from office.

I discovered a pattern of corruption that the Washington press has been covering up for three or four years Rudy Giuliani

Parnas and Fruman were arrested last month on a four-count indictment that includes charges of conspiracy, making false statements to the Federal Election Commission and falsification of records. Both pleaded not guilty.

Parnas has indicated a willingness to cooperate with Congress. On Friday, CNN reported that a lawyer for Parnas said his client was willing to testify about meetings between Devin Nunes, the top Republican on the House intelligence committee conducting the impeachment inquiry, and a former Ukrainian prosecutor general in Vienna in 2018, allegedly to discuss digging up dirt on Biden.

The same evening, newly released state department documents showed Parnas and Fruman’s involvement in contacts between Giuliani and the secretary of state, Mike Pompeo, in the months before the US ambassador to Ukraine was abruptly recalled, an event under scrutiny by the House panel.

The documents were released to the group American Oversight in response to a Freedom of Information Act lawsuit. They showed that Pompeo talked to Giuliani on 26 March and 29 March. Last week, former ambassador Marie Yovanovitch told House investigators she felt “kneecapped” by a “smear campaign” Giuliani led against her. She was withdrawn from Ukraine in May.

The documents also included a report that appeared with Trump hotel stationery and seemed to summarize a 23 January interview with former Ukrainian prosecutor general Viktor Shokin at which Giuliani, Parnas and Fruman were present.

The CNN report about Nunes’ trip to Europe quoted the attorney Joseph A Bondy as saying Parnas was told by Shokin that he met the California Republican in Vienna last December. Nunes responded by telling the far-right Breitbart News website he would sue CNN and the Daily Beast, which reported on the same subject.

A second state department memo released on Friday appeared to summarize an interview with Yuri Lutsenko, another former prosecutor general of Ukraine, conducted in the presence of Giuliani, Parnas and Fruman. Lutsenko was quoted as raising questions about compensation Joe Biden’s son, Hunter Biden, received from the Ukrainian energy company Burisma.

There is no evidence either Hunter Biden or his father committed any wrongdoing related to Ukraine.

Speaking to Fox News on Saturday, Giuliani was asked about Parnas and Fruman.

“So they helped me find people,” he said, “and as I’ve said, they did a good job, but they weren’t investigators, and they weren’t James Bond, and they didn’t have personal communications with the president.”

Giuliani admitted introducing the men to Trump at a Hanukkah party in December 2018 – the subject of another CNN report – but said there was no extended conversation.

“They took a one-minute picture,” he said. “They walked away.”

CNN cited sources as saying Parnas said “the big guy”, meaning Trump, had “talked about tasking him and Fruman with what Parnas described as ‘a secret mission’ to pressure the Ukrainian government to investigate” the Bidens.

Giuliani himself is being investigated by federal prosecutors in New York, over whether he failed to register as a foreign agent. On Fox News, the former mayor was asked if he was concerned about being indicted.

“Do you think I’m afraid?” he said. “Do you think I get afraid? I did the right thing. I represented my client in a very, very effective way. I was so effective that I discovered a pattern of corruption that the Washington press has been covering up for three or four years.”

Giuliani has defied attempts to compel him to turn over documentation relevant to the impeachment inquiry. He has also appeared dramatically in public testimony, described as being involved in a political “drug deal” and as “a hand grenade that was going to blow everyone up”.

On Saturday he released a dramatic letter to the South Carolina senator Lindsey Graham, a key Trump ally, in which he claimed witnesses damaging to House Democrats’ case against Trump were being kept from coming to the US from Ukraine.

Giuliani told Fox he continues to have a good relationship with Trump, to whom he talks “early and often”. He also said he had seen it written that Trump was intending to throw him under the bus.

“When they say that, I say he isn’t, but I have insurance,” Giuliani said.

That was a repeat of a remark made to the Guardian, prompting his own lawyer to interject: “He’s joking.”

Later on Saturday, Giuliani returned to the subject on Twitter, writing that his remark was “sarcastic [and] relates to the files in my safe about the Biden Family’s [four-] decade monetizing of his office. If I disappear, it will appear immediately”.