The administration reportedly wants CIA Director Mike Pompeo to replace Tillerson.
One of the top Republicans is becoming even more outspoken against Trump after announcing his retirement.
Tillerson also wants to remove an envoy for the closure of Guantánamo Bay.
America's top diplomat is "entitled to take a few days himself," said a State Department spokeswoman.
There is still "plenty of work to do" to restructure the State Department, Tillerson's spokesman says.
The 22-year-old University of Virginia student was sentenced to 15 years' hard labor.
The U.S. Secretary of State faces a tough negotiation in his Moscow visit after accusations of Russian complicity in Syrian chemical attacks.
Tillerson's potential no-show had increased unease caused by U.S. President Donald Trump's description of NATO as "obsolete" during his election campaign.
The secretary of state and President Trump have been critical of China in recent weeks.
Within hours of starting the job, Tillerson reached out by phone to counterparts from Mexico and Canada, and met with German Foreign Minister Sigmar Gabriel, department officials said.
Despite concerns about the former Exxon Mobil CEO's ties to Russia, he won a 53-42 vote.
Under Secretary for Management Patrick Kennedy and acting Under Secretary of State for Arms Control and International Security Thomas Countryman both plan to leave their posts by Friday, the officials said.
The U.S. Senate Foreign Relations Committee voted 11-10 to approve Tillerson, with every committee Republican backing Tillerson and every Democrat opposing his nomination.
He did a deal with the Kurds, cozied up to African despots and befriended Putin.
Some in Russia are lauding Trump's nomination of the Exxon CEO for secretary of state, although opponents say it 'would take a miracle for Tillerson to criticize Russia on human rights.'
Republicans and Democrats said Tillerson would be asked about his ties to Russia.
'Additional candidates' to be interviewed this week as the field widens beyond Romney, Giuliani, Petraeus, Corker.
Former Trump campaign manager blitzes the Sunday news shows with message that Romney cannot be the choice for Secretary of State.