Supporters of jailed Venezuelan opposition leader Leopoldo Lopez have held a vigil outside his prison demanding to see him after rumours about his health rattled the protest-hit country.

Lopez's wife and mother rushed to a military hospital in Caracas and then the hilltop Ramo Verde jail overnight to Thursday, after a journalist tweeted Lopez had been taken to a medical centre without vital signs.

President Nicolas Maduro's leftist government, facing a wave of major opposition protests since last month, later issued a short "proof of life" video in which Lopez said he was fine.

Officials accused Lopez's family of stirring up a media frenzy to get attention and further stoke the protests.

"Today is May 3, it's 9pm. ... I'm sending a message to my family and my children that I am well," said Lopez, 46, standing cross-armed in front of cell bars and looking healthy in a sleeveless white T-shirt.

But Lopez's wife Lilian Tintori, who says she has not been allowed to see him in over a month, rejected the video as "false" and spent the night outside the jail.

"The only proof of life that we will accept is to see Leopoldo," she tweeted, as she and Lopez's mother faced a line of green-clad National Guard soldiers at the prison. They later rotated out with some supporters to get sleep.

Fresh protests

Protests continued on Thursday, in the capital Caracas and elsewhere.

Gunfire erupted at a student gathering in El Tigre, a city southeast of Caracas, leaving Juan Lopez, 33, dead and three others injured, according to the chief prosecutor's office.

Preliminary reports indicate an assailant fired at the student leader towards the end of the meeting and then fled on a motorcycle.

His death brought to at least 37 the number killed in Venezuela's ongoing political turmoil, including protesters, government supporters, bystanders and security officials.

READ MORE: Venezuela's crisis, explained

Earlier on Thursday, authorities announced a 38-year-old police officer in the central state of Carabobo had died of his injuries after being shot during a Wednesday protest that had hundreds of thousands of people on the street nationwide.

Venezuelans were shaken by the rumours over Leopoldo Lopez,who is Venezuela's most prominent imprisoned politician. He was jailed in 2014 during the last major round of protests.

The US-educated economist and leader of the hard-line Popular Will party is accused of inciting violence, and in 2015 was sentenced to almost 14 years behind bars.

The government says he is a dangerous agitator, pointing to his involvement in a brief 2002 coup against the late Hugo Chavez, when Lopez even helped arrest a cabinet minister.

"They're inventing that something or other has been done to Leopoldo to put together a big, pretty show, so that we forget the 43 deaths he caused," said Socialist Party No. 2 Diosdado Cabello, in reference to those killed during unrest in 2014.