Tillerson's comments to reporters came during a visit to Geneva in which he met UN envoy Staffan de Mistura, who is trying to convene a new round of peace talks next month.
The secretary said US policy has not changed, but his remarks represented tougher language from an administration that had previously said Assad's fate is not a priority.
"We do not believe there is a future for the Assad regime, the Assad family," Tillerson said.
"I think I've said it on a number of occasions. The reign of the Assad family is coming to an end, and the only issue is how should that be brought about."
De Mistura hopes to convene an eighth round of Syrian peace talks between Assad's regime and an opposition coalition in Geneva from November 28.
These will be focused on drafting a new constitution and holding UN-supervised elections in a country devastated by several overlapping bloody civil conflicts.
Assad's regime has been saved by Russian and Iranian military intervention and he insists that he will not stand down in the face of what he regards as "terrorist" rebels.
But Western capitals, the opposition and many of Syria's Arab neighbors hold Assad's forces responsible for the bulk of the 330,000 people who have died in the conflict.
He has been accused of using chemical weapons against his own people and of overseeing the large-scale torture and murder of civilian detainees.
The previous US administration often said that Assad's days were numbered, but the then president Barack Obama decided not to use force to punish his chemical weapons attacks.
His successor, President Donald Trump, did order one missile strike on a Syran air base in response to a chemical attack.
But US policy has otherwise focused solely on the defeat of the Islamic State jihadist group, driving it out of its last bastions in eastern Syria's Euphrates valley.
Tillerson said, however, that he hopes a way to oust Assad will "emerge" as part of De Mistura's UN-mediated talks.
He argued that the UN Security Council resolution setting up the peace process already contains a procedure to hold elections that Washington does not think Assad can win.
"The only thing that changed is when this administration came into office, we took a view that it is not a prerequisite that Assad go before that process starts, rather the mechanism by which Assad departs will likely emerge from that process," he said.
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