The source said he had heard nothing about any possible departure, but added: "The situation doesn't seem to be getting any better, and in some respects appears to be getting worse."
R.C. Hammond, Tillerson's spokesman, denied Tillerson was considering leaving or that his frustrations were boiling over, saying he had "plenty of reasons to stay on the job, and all of them are important to America."
"There's a desperate need for American leadership in the world and that's where the secretary's focusing his attention," he said.
Tillerson scored a policy win last week when the administration certified, albeit reluctantly, that Iran was complying with the 2015 nuclear deal under which Tehran agreed to restrain its atomic program in exchange for sanctions relief.
He was upset, however, by fierce internal criticism from Trump, as well as his chief strategist, Steve Bannon, and White House aide Sebastian Gorka, over the decision, said another U.S. official who spoke on condition of anonymity.
"The secretary does not feel that White House staff members should be in a position to conduct hostile cross-(examinations) of Cabinet officials," the official said.
Hammond disputed the account of harsh discord between Trump and Tillerson regarding recertifying the Iran nuclear deal, saying: "I don't buy this whole thing that there are tensions. Developing public policy is about vetting out ideas," he said.
Following last month's move by Saudi Arabia, the United Arab Emirates, Bahrain and Egypt to boycott Qatar, which they accuse of financing extremist groups and supporting terrorism, Tillerson publicly asked the nations to ease their blockade, and put the onus on both sides to end the crisis.
Less than 90 minutes later, Trump accused Qatar of being a "high level" sponsor of terrorism and suggested he had helped plan the Qatar action with Arab leaders.
"As a government we offered some confusing signals, but we've since been able to clear up and clarify that," Hammond said.
Politico reported last month that Tillerson was furious with Johnny DeStefano, the head of the presidential personnel office, for torpedoing proposed nominees to key State Department posts.