Washington (CNN) A source close to President Donald Trump's legal team tells CNN that former New York Mayor Rudy Giuliani is still the President's attorney but will not be dealing with matters involving Ukraine.

Earlier on Friday, Trump wouldn't say whether Giuliani was still his personal attorney.

"Well, I don't know. I haven't spoken to Rudy. I spoke to him yesterday briefly. He's a very good attorney and he has been my attorney, yeah, sure," he said.

When asked later by CNN if he was still Trump's attorney, Giuliani responded, "Yes."

He added, "There are no Ukraine issues. I finished that in March. I'm still representing him."

The skepticism, which is shared by many of Trump's allies, comes after two of Giuliani's clients were arrested boarding an international flight and charged for violating campaign finance laws.

As the drama unfolded on television -- complete with scowling mug shots paired with photos of the two men posing alongside the President -- Trump began expressing concerns about Giuliani's involvement with the individuals, according to people familiar with the matter.

The White House was informed of the arrest on Wednesday night, according to a law enforcement official. The development lends a further sense of uncertainty to a White House legal strategy which appears, after only three days, to be unsuccessful in preventing administration officials from cooperating with Congress in its impeachment probe.

Efforts to displace Giuliani with a new legal mouthpiece appeared to hit a snag on Thursday. A day after Trey Gowdy, the former South Carolina congressman, was announced as part of his legal team, Trump unexpectedly declared he wouldn't be able to start until "sometime after January."

The decision came after legal concerns about lobbying within the year after leaving elected office. The law stipulates former congressional members cannot lobby with the intent of impacting decisions made by current members within a year of leaving office. Gowdy left Congress in January 2019.

The announcement prompted a sigh of relief from one corner of the West Wing who were strongly opposed to Trump bringing Gowdy on to the team. Other Trump allies questioned both Gowdy's loyalty to the President and his legal acumen, pointing to his handling of the congressional probe into the 2012 attack on the Benghazi, Libya, diplomatic facility and his advice to Trump that he cooperate with the Mueller investigation.

"Trey Gowdy doesn't know s***," Victoria Toensing, a lawyer who has worked with Giuliani and is close to Trump, told Yahoo News.

But those voices were counterbalanced by other aides and some of Trump's allies outside the White House, who have warned Trump for several weeks that Giuliani would not act as a sufficient surrogate during what is turning into a brutal impeachment fight.

Still Trump's attorney

There are growing concerns in Trump's orbit that Giuliani is increasingly becoming a political and potentially legal liability, even before two of his clients were arrested.

These aides have told Trump that Giuliani was only damaging his defense -- and after Thursday's arrests, Trump has raised his own questions privately about Giuliani's culpability, according to people familiar with his concerns.

The White House declined to comment on Trump's position on Giuliani.

Giuliani's financial dealings with the two associates indicted on campaign finance-related charges are under scrutiny by investigators overseeing the case, law enforcement officials briefed on the matter said. The FBI and prosecutors in Manhattan are examining Giuliani's involvement in the broader flow of money that have become the focus of alleged violations that are at the center of the allegations.

Asked by CNN Thursday if he feared his personal attorney could be indicted potentially, Trump said: "I hope not."

He then put distance between himself and the actions undertaken by his lawyer.

"You know, he's got a lot of clients. So, I just don't know. I haven't spoken to Rudy about it, I don't know," Trump said.

Trump has shown little apprehension in the past at unceremoniously cutting ties with onetime allies when they find themselves in legal hot water. That includes his attempts to distance himself from another personal attorney, Michael Cohen, when it became clear Cohen was in legal trouble for arranging payments during the 2016 election to silence women who claimed affairs with Trump.

Trump wrote-off Cohen -- a longtime fixer and aide -- as only responsible for "low-level work" the day after he was sentenced to three years in prison.

Still, Trump's relationship with Giuliani runs deeper than most of his current political alliances. They rose to prominence together in 1980s and 90s New York, supported each others' political endeavors and -- when they failed -- offered consolation (Giuliani has said he spent a month at Trump's Mar-a-Lago estate after an ill-fated presidential run in 2008).

For those reasons, some people close to Trump say tossing Giuliani aside would prove more difficult for him emotionally than any of the past advisers who were summarily discarded. And fundamentally, Trump believes Giuliani is working in his best interests without an alternative agenda.

There's also the question of what dismissing Giuliani would actually mean. He is not being paid for his services to Trump, and does not receive marching orders from anyone beyond the President himself.

Looking for a change

Aides inside the White House are continuing to search for other options that would provide a strong television presence that might supplement -- or supplant -- Giuliani's often freewheeling appearances that many of Trump's associates do not view as helpful.

The move to bring on Gowdy had been in the works for several weeks, and was viewed as a way to change directions from Giuliani, one source said. He hasn't been assigned specific tasks and -- if the legal hurdles to his employment are cleared -- could take the place of Giuliani on television and play an "offensive role," according to a person familiar with the matter.

Trump's attorneys are also looking to hire more lawyers to join the team headed by Jay Sekulow, who has represented Trump since the Mueller investigation. Jane and Marty Raskin, who run a well-regarded white collar defense firm in South Florida and formed part of Trump's Russia legal team, are also providing legal assistance, though one source said they never ended their ties to Trump when the Mueller investigation concluded.

The Gowdy delay is a blow to Mick Mulvaney, the acting chief of staff, who pushed for his friend from South Carolina to join the team. Mulvaney has adopted a lower profile during the impeachment battle, leaving another adviser -- Jared Kushner, Trump's son-in-law -- as the de-facto boss.

Trump had initially dismissed the idea of bringing on additional attorneys, but blessed Gowdy's hiring after a long Oval Office meeting with him this week. Now Mulvaney finds himself without a reliable ally on the outside defense team.

Mulvaney has been at odds with the chief counsel inside the White House for several days, feuding with Pat Cipollone over the decision to publish an eight-page letter calling the impeachment inquiry illegitimate and declaring the White House will not cooperate unless until the House votes to open an investigation.

Strength tested

The strength of that legal strategy is being tested by a roster of administration officials due to appear before Congress.

In their letter to congressional Democrats, White House lawyers wrote: "President Trump and his Administration cannot participate in your partisan and unconstitutional inquiry."

But already a number of administration officials have signaled they are willing to break with Trump's dictate to not cooperate in the investigation.

On Friday, the ousted US Ambassador to Ukraine Marie Yovanovitch -- still a State Department employee -- sat for hours of deposition with the House Intelligence Committee. As she was testifying, Democrats said the White House had attempted to block her appearance the prior evening, forcing them to issue a subpoena.

And after his voluntary appearance was derailed by the State Department this week, US Ambassador to the European Union Gordon Sondland now plans to appear next Thursday after being subpoenaed by congressional investigators.

Democrats have also scheduled depositions next week with Deputy Assistant Secretary of State George Kent and State Department counselor Ulrich Brechbuhl. Fiona Hill, the White House's top Russia adviser who left the administration in August, has been scheduled to appear for a deposition on Monday.

It's not clear if all intend to appear, and one Trump congressional ally downplayed the testimony of administration officials as defiance of the White House, saying they're only doing so under subpoena.

But the pattern of administration officials choosing Congress over the White House remains a remarkable choice for a President who routinely demands loyalty from those who work for him.

CORRECTION: This story has been updated to correct the legal concerns around Trey Gowdy joining Trump's legal team.