Trump met Tillerson on Tuesday and may talk to him again over the weekend, the official said. Trump appears to be in the final days of deliberations over his top diplomat with an announcement possible next week.
Tillerson's favoured status was revealed as former New York Mayor Rudy Giuliani formally withdrew from consideration for secretary of state.
The transition official, who spoke on condition of anonymity, said Tillerson, 64, had moved ahead in Trump's deliberations over 2012 Republican presidential nominee Mitt Romney, who has met Trump twice, including at a dinner in New York.
But the official said Romney was still under consideration for the job, along with John Bolton, a former US ambassador to the United Nations; US Senator Bob Corker of Tennessee, and retired Navy Admiral James Stavridis.
Giuliani's withdrawal came after he was fully vetted by the Trump transition team for his overseas business ties in what was described by the Trump official as an "intense" effort by lawyers and accountants.
Giuliani, who runs a global consulting firm, was given a clean bill of health, with Trump's aides concluding his business interests would not pose a risk to his confirmation.
Should Tillerson be nominated, his business ties, too, will come under scrutiny. Exxon Mobil has operations in more than 50 countries and boasts that it explores for oil and natural gas on six continents.
In 2011, Exxon Mobil signed a deal with Rosneft, Russia's largest state-owned oil company, for joint oil exploration and production. Since then, the companies have formed 10 joint ventures for projects in Russia.
In 2013, Russian President Vladimir Putin awarded Tillerson his nation's Order of Friendship.
But US sanctions against Russia for its incursion into Crimea cost Exxon Mobil dearly, forcing it to scrap some projects and costing it at least $1 billion in losses. Tillerson has been a vocal critic of the sanctions.
Trump has spoken of wanting warmer relations with Moscow, which has sparked concerns in Congress that he could lift or loosen some of the sanctions on Russia.
Tillerson has been chairman and CEO of Exxon Mobil since 2006. He is expected to retire from the company next year.
Should Tillerson be nominated, climate change could be another divisive issue. The company is under investigation by the New York Attorney General's Office for allegedly misleading investors, regulators and the public on what it knew about global warming.