Tillerson told anchor Jake Tapper that he maintains a close and candid relationship with Trump, despite reports he has been frustrated with the commander in chief and indications that he remains on thin ice with the administration.
He also repeatedly refused to say whether he called Trump a "moron"
following a meeting at the Pentagon this past summer.
"Jake, as I indicated earlier when I was asked about that, I'm not going to deal with that kind of petty stuff," Tillerson said. "I mean, this is a town that seems to relish in gossip, rumor and innuendo. And they feed on it. They feed on it in a very destructive way. I don't work that way and I don't feel that way."
Asked about Tennessee Sen. Bob Corker's comments suggesting President Donald Trump's tweets and foreign policy comments "castrate" the secretary's diplomatic efforts, Tillerson said simply: "I checked ... I'm fully intact."
Corker told The Washington Post
in an interview last week that Trump's public statements on foreign policy undercut his secretary of state -- creating "binary" scenarios for the United States on the world stage.
"You cannot publicly castrate your own secretary of state without giving yourself that binary choice," Corker told The Washington Post's Jackson Diehl in a phone interview on Friday. "The tweets -- yes, you raise tension in the region (and) it's very irresponsible."
Iran nuclear deal
Tillerson said Sunday the US is trying to stay in the Iran nuclear deal while hoping to achieve more from it, days after Trump threatened to pull the US out of the agreement.
"We're going to stay in," Tillerson said, though he left open the possibility that the US could seek another agreement.
"We're going to work with our European partners and allies to see if we can't address these concerns," he added.
Trump last week said Iran was violating the agreement and threatened to pull out of the deal, pushing the issue to Congress. Tillerson has said in the past Iran was in "technical compliance" with the deal.
Asked to square these two claims, Tillerson said Iran had committed technical violations of the deal, but that the agreement was structured in a way to give Iran significant time to remedy its violations.
On North Korea, Tillerson said his diplomatic efforts will continue
as long as they possibly can despite the saber-rattling on both sides of the Pacific.
"Those diplomatic efforts will continue until the first bomb drops," Tillerson said.
Tillerson's comments came a day before the latest US-South Korea military exercises off the Korean Peninsula, which Pyongyang claims are rehearsals for a US-led invasion. The US says the drills are defensive in nature.
Tillerson said prioritizing diplomacy to try to curb North Korea's nuclear ambitions is President Donald Trump's preferred avenue. Despite a statement on Twitter that suggested diplomacy won't work, the President wants to avoid violence, Tillerson said.
"The President has also made clear to me that he wants this solved diplomatically," Tillerson said. "He is not seeking to go to war."