Tillerson, making his first trip to Africa as the senior-most United States diplomat, has visited Ethiopia, Djibouti and Kenya since last Wednesday.
Mr Tillerson was to hold an interactive meeting with Health CS Sicily Kariuki at the Pumwani Maternity Hospital on Saturday at 9am but the event did not take place.
Tillerson, 65, is four days into his first diplomatic trip to Africa, where he has visited Ethiopia, Djibouti and Kenya, and is scheduled to go on to Nigeria and Chad.
Tillerson, who was scheduled to visit Nigeria would return to Washington, a day ahead of his scheduled visit to the country.
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According to a report on Reuters, Tillerson would now leave Africa to arrive U.S. early Tuesday morning as against his previous plan of returning on Tuesday night. He also planned to meet US Embassy staff, and attend a working lunch with Monica Juma, the nation's minister for foreign affairs.
Tillerson was in Chad on Monday for talks with President Idriss Deby, an important American ally whose military is battling al-Qaeda and ISIS-affiliated groups in the Sahel region.
"In particular, he plans to discuss ways we can work with our partners to counter terrorism, advance peace and security, promote good governance, and spur mutually beneficial trade and investment", Nauert said.
But the African tour has coincided with the shock announcement of a possible summit between North Korean leader Kim Jong Un and US President Donald Trump over the nuclear standoff in the Korean peninsula.