The businessman, Reza Zarrab, had ties to the top Turkish government officials, who had driven the US to make allegations that Zarrab had violated US sanctions on Iran's nuclear program. He was a customer of Giulianis.
During the White House meeting, President Giuliani and Michael Mukasey, the former attorney, offered to pitch to Tillerson, the source said.
After the meeting, Tillerson mentioned the discussion to the then Chief of Staff John Kelly, who was not present at the meeting and noted that it was a deal he could not support.
This section is another reflection of Trump trying to use his office to influence ongoing Justice Department procedures. And news of that comes when his relationship with both Giuliani and Turkish President Recep Tayyip Erdoğan comes under scrutiny.
Giuliani proposed swapping Zarrab for a US pastor held by Turkey, Andrew Brunson. Asked by the president if the deal was one he could support, Tillerson said he couldn't intervene and cited the ongoing Justice Department case.
The President asked Tillerson if the affair was something Jeff Sessions, then Attorney General, could do. Tillerson replied that Sessions would have to contend with the fact that it was an ongoing case on the way to the trial. He also told the president that it would be bad policy because it could jeopardize other Americans and raise the prospect that more hostages could be used as leverage for similar deals.
Mukasey, who worked with Giuliani to represent Zarrab, told the president that it was an arrangement he could order as president because it was within his power. Trump told Tillerson and Giuliani to see if they could resolve an arrangement.
Tillerson said he couldn't support it.
Giuliani pursued the possible agreement directly with Tillerson after the Oval Office meeting and was rejected, the source said. Giuliani and Mukasey also made a similar pitch to Justice Ministry officials, who opposed, according to sources informed of these discussions.
Mukasey, reached Thursday, said he would not discuss the White House meeting and declined comment. The White House did not respond to a request for comment. Tillerson did not immediately respond to a request for comment.
Giuliani initially denied the episode to Bloomberg, but later he told the outlet that it was possible that it was true.
"Suppose I talked to Trump about it -" So what? I was a private lawyer at the time, "Giuliani told Bloomberg. "Maybe I'm wrong. Maybe I somehow lost his name in a conversation. Or maybe one of his people talked to him about it because I was trying to make a prisoner change."
The revelation comes as Democrats investigate Trump's ties to Giuliani, who played a key role in efforts to secure dirt on Joe Biden from Ukraine. Trump's own phone call with the President of Ukraine asking him to investigate Biden is central to ongoing impeachment efforts.
At the same time, Trump's relationship with Erodgan has received renewed attention after he spoke with the authoritarian leader over the weekend and abruptly announced that US troops would withdraw from northern Syria to clear the path for a Turkish invasion.
Since leaving the Department of Foreign Affairs in 2018, Tillerson has spoken of Trump's habit of asking him to make things unconscious that such measures would violate the law.
"When the President would say," Here's what I want to do and here's how I want to do it. "And I have to say to him," Well, Mr. President, I understand what you want to do, but you can't do it that way. It violates the law, "he said during a gathering in December.  Tillerson said Trump would be" really frustrated "when he told him his inquiries would be illegal.