Todd responded, "Truth is truth."
"No, it isn't truth. Truth isn't truth," Giuliani said.
The exchange was heavily featured on news shows, with pundits comparing Giuliani's assertion with White House adviser Kellyanne Conway saying earlier during Trump's tenure that then-press secretary Sean Spicer had presented "alternative facts."
Giuliani has said he fears Mueller would be more inclined to rely on testimony provided by former FBI Director James Comey, whom Trump fired, than that of the President. If so, Giuliani has said, Trump could be accused of perjury even if he's telling the truth.
Following Giuliani's appearance, Comey took to Twitter, writing: "Truth exists and truth matters."
"Truth has always been the touchstone of our country's justice system and political life," Comey said. " People who lie are held accountable. If we are untethered to truth, our justice system cannot function and a society based on the rule of law dissolves."
Today, Giuliani responded to Comey.
"Since proven leaker Comey is now sanctimoniously lecturing us on truth, I can say sometimes the truth is clear it's the opposite of what Jim Coney is saying," Giuliani wrote on Twitter.
"His disgraceful performance with the FBI makes him the last person who should pontificate on truth."