Mr Maduro was speaking at an military event in Caracas when the alleged attack occurred.
Live footage of Mr Maduro’s speech shows the president suddenly looking upwards – startled – and dozens of soldiers running away.
Mr Maduro has blamed Colombia for the attack – something denied by Bogota as a “baseless” accusation.
Seven soldiers were injured, and several people were later arrested, the Venezuelan authorities said.
The incident happened when Mr Maduro was speaking at the event to mark the 81st anniversary of the national army.
Two drones loaded with explosives went off near the president’s stand, Communications Minister Jorge Rodriguez said.
Mr Maduro later said in a national address: “A flying object exploded near me, a big explosion. Seconds later there was a second explosion.”
Photos on social media showed bodyguards protecting Mr Maduro with bulletproof shields after the alleged attack.
Mr Maduro accused neighbouring Colombia and elements within the US of instigating “a right-wing plot” to kill him.
He added that he had “no doubt” Colombian President Juan Manuel Santos was “behind this attack”.
The Venezuelan leader, who has previously accused the US of plotting against him, provided no evidence to back his claim.
The Colombian government has denied any involvement, saying there is “no basis” to Mr Maduro’s allegations.
By contrast, Mr Rodriguez accused Venezuela’s right-wing opposition of carrying out the attack.
“After losing the vote, they failed again,” Mr Rodriguez said.
He was referring to May’s presidential elections, where Mr Maduro was re-elected for another six-year term.
However, Hasler Inglesias, a youth leader with the opposition Voluntad Popular Party, told the BBC: “We didn’t know what was happening. It’s hard to believe that the opposition is going to make an attempt when they have never made an attempt in this way in 20 years.”
Meanwhile, a little-known group called Soldiers in T-shirts said on social media that it was behind the alleged attack.
It said it had planned to fly two explosives-laden drones at Mr Maduro, but they were shot down by the military.
The claim was not backed up by any evidence, and the group did not respond to media requests for comment.
To add further to the confusion, firefighters at the scene disputed the government’s version of events, the Associated Press reports.
Speaking on condition of anonymity, three of them said the incident was actually a gas tank explosion inside an apartment, but did not provide further details, the news agency says. (Courtesy BBC)