“Picking exact days is difficult. I’m confident that the Venezuelan people will ensure that Maduro’s days are numbered,” Mr Pompeo told CNN.
Two people died in Saturday’s clashes between civilians and troops loyal to Mr Maduro, who blocks aid deliveries.
Self-declared interim President Juan Guaidó said Mr Maduro must resign.
Mr Guaidó also called on other nations to consider “all measures” to oust Mr Maduro after opposition-led efforts to bring in aid descended in the clashes.
Mr Guaidó marshalled volunteers to collect and transport the aid from Brazil and Colombia – but this set off fierce border clashes with soldiers, who opened fire using a mixture of live ammunition and rubber bullets.
US President Donald Trump says he has not ruled out an armed response.
On Sunday, the European Union joined the condemnation.
“We reject the use of irregular armed groups to intimidate civilians and lawmakers who have mobilised to distribute aid,” said EU foreign policy chief Federica Mogherini.
Mr Guaidó last month declared himself interim president and has since been recognised by more than 50 countries.
He argues that alleged irregularities with the nation’s 2018 election render Mr Maduro’s leadership illegitimate.
Mr Maduro, who is backed by key economic allies including Russia, Cuba and China, says he is the legitimate president.
He also warns that deliveries of foreign aid would open the way for US military intervention.
What happened at the border on Saturday?
Led by Mr Guaidó, Venezuela’s opposition had intended to peacefully bring aid trucks over the borders with Brazil and Colombia. Soaring inflation has left many Venezuelans unable to afford basic items such as food, toiletries, and medicine.
Mr Guaidó had pledged that the aid would come into the country on Saturday. In response, Mr Maduro partly closed the country’s borders, citing threats to security and sovereignty.
Venezuelans civilians attempted to cross in order to get to the stores of food and medicine, but the attempt quickly descended into bloody violence along Venezuela’s southern border with Brazil and western border with Colombia.
At least 60 soldiers had defected by late Saturday, according to Colombia’s migration service, but most of the military appeared to still be loyal to Mr Maduro.
Video footage showed Venezuelan soldiers crashing their armoured vehicles into the border with Colombia in order to defect.
Another video posted on social media appeared to show four soldiers publicly denouncing the president and announcing their support for Guaidó. “We are fathers and sons, we have had enough of so much uncertainty and injustice,” the soldiers said.
Mr Guaidó promised the defectors amnesty if they joined the “right side of history”.
At about 19:00 local time (23:00 GMT) on Saturday, Colombia’s government estimated the number of injured at border crossings to be about 300.
Journalists at the scene reported severe injuries among protesters, including several who appeared to have lost their eyes.
Amnesty International described the use of live ammunition against protesters as a crime under international law.
How has Maduro reacted?
President Maduro has ignored international calls to hold new elections.
He has accused Mr Guaidó of being a “puppet”, an “American pawn”, a “clown” and an “imperialist beggar”.
As protests got under way at Venezuela’s borders, Mr Maduro staged a rally in Caracas.
“Take your hands off Venezuela, Donald Trump,” he told a cheering crowd.
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The US is leading the international effort to pressure Mr Maduro, and has implemented a raft of financial sanctions against his government.
How did we get to this point?
The humanitarian aid stockpiled in Colombia and Brazil is at the centre of a stand-off between Mr Maduro and Mr Guaidó that goes back to Mr Maduro’s 2018 re-election – a vote Mr Guaidó declared illegitimate.
For several years Venezuela has been in the grip of a political and economic crisis. An out-of-control inflation rate has seen prices soar, leaving many Venezuelans struggling to afford basic items.