"I will be making my announcement on the next Secretary of State tomorrow morning," Trump tweeted.
However, soon after that former Massachusetts Governor Mitt Romney, who was one of the leading contenders for the post, said he would not be the Secretary of State.
"It was an honour to have been considered for Secretary of State of our great country. My discussions with President- elect Trump have been both enjoyable and enlightening. I have very high hopes that the new administration will lead the nation to greater strength, prosperity and peace," Romney said.
Tillerson is a veteran chief executive who has had extensive overseas business dealings but his relationships with foreign leaders could complicate his confirmation prospects, the daily said.
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If confirmed by the Senate, Tillerson would replace John Kerry as the Secretary of State. Over the weekend, Trump had praised Tillerson.
"He's more than a business executive; he's a world-class player. He's in charge of I guess the largest company in the world," Trump had said.
"To me, a great advantage is he knows many of the players, and he knows them well. He does massive deals in Russia. He does massive deals for the company -- not for himself, for the company," Trump said.
"The nomination of ExxonMobil chief executive Rex Tillerson, which was confirmed by a person involved with the transition, likely faces a tough fight in the Senate, where some Republicans have said they are worried about his relationship with Russian President Vladimir Putin," The Washington Post reported.
In addition to Romney, other names in consideration for the position were the Senate Foreign Relations Chairman Bob Corker and the former CIA Director Gen (rtd) David Petraeus.
One of Trump's closest friend Rudy Giuliani, the former New York mayor, had withdrawn his name from being considered for any position in the Trump Administration.
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"Nominating Exxon CEO Rex Tillerson to be Secretary of State would be handing over the keys to US foreign policy to Big Oil. Donald Trump would be sending a strong signal to OPEC and the entire global fossil fuel industry that oil will be America's primary foreign policy priority. We cannot allow oil to replace diplomacy as the currency of the State Department," Senator Edward Markey said in a statement.
"This nomination is clear confirmation that a Trump administration will doggedly pursue an 'oil above all' energy agenda, at the expense of clean energy and climate solutions," he said.
"The role of Secretary of State should not be bestowed upon someone whose only notable experience with foreign governments involve multi-million dollar deals with Russia and whose experience with the federal government seems to be limited to campaigns against the effects of climate change," demanded Senator Robert Menendez.
"For the world after 2008, and globalisation, both its benefits and it's ills, is in dire need of real sustained recovery that every hard-working American can enjoy while every other person across the world enjoys improved human rights," he said.
"Rex Tillerson'sinnate ability to remain sharply focused on the prize, aided by mastery of the nuance and a warm comfortable demeanor will represent Trump's resurgent and confident America to everyone on earth: friend and foe alike. Only a confident America can deliver enhanced global peace and security for all," Batra said.