Chavez has not been seen or heard from since the Dec. 11 operation, and officials have reported a series of ups and downs in his recovery the most recent, late Sunday, announcing that a respiratory infection had put the president in a "delicate" state.
Jorge Rodriguez, a Chavez ally and mayor of a Caracas district, reiterated on Tuesday that the president is going through a "complex post-operative process."
He told reporters that Venezuelans have shown an outpouring of compassion and support for a leader who has "been planted in the hearts of millions." Rodriguez urged Venezuelans to keep Chavez in their prayers and expressed hope the president would recover.
Chavez's son-in-law Jorge Arreaza, who is the government's science minister and has been with the president in Cuba, urged Venezuelans in a Twitter message Monday night not to believe "bad-intentioned rumors" circulating online. "President Chavez has spent the day calm and stable, accompanied by his children," Arreaza said in the message.
His tweet came a day after Vice President Nicolas Maduro's grim statement from Havana that Chavez had suffered new complications due to a respiratory infection. Maduro had said last week, before seeing Chavez, that the president had been up and walking.
Political opponents of the socialist Chavez have complained that the government hasn't told the country enough about his health, and even some of his supporters said Tuesday that they wished they knew more.
"We're distressed by El Comandante's health," said Francisca Fuentes, who was walking through a downtown square with her grandchildren. "I think they aren't telling us the whole truth. It's time for them to speak clearly. It's like when you have a sick relative and the doctor lies to you every once in a while."
Chavez has been fighting an undisclosed type of pelvic cancer since June 2011. He has declined to reveal the precise location of the tumors that have been surgically removed. The president announced on Dec. 8 that his cancer had come back despite previous surgeries, chemotherapy and radiation treatment.
The Venezuelan government has not given details about Chavez's latest complications beyond saying that they arose from the respiratory infection.
"There's nothing we can do except wait for the government to deign to say how he is really," said Daniel Jimenez, an opposition supporter who was in a square in an affluent Caracas neighborhood.
Jimenez and many other Venezuelans say it seems increasingly unlikely that Chavez can be sworn in as scheduled Jan. 10 for his new term.
Venezuelans rang in 2013 as usual with fireworks raining down all over the capital of Caracas. But some of Chavez's supporters had long faces as they gathered in Bolivar Plaza on Monday night holding pictures of the president. A government-sponsored New Year's Eve celebration there had been called off.
Chavez's supporters instead strummed guitars and read poetry in his honor in the plaza, singing along with a recording of the president belting out the national anthem.
Chavez has been in office since 1999 and was re-elected in October, three months after he announced that his latest tests showed him to be cancer-free. If he dies or is unable to continue in office, the Venezuelan Constitution says a new election should be held within 30 days.
Before his operation, Chavez acknowledged he faced risks and designated Maduro as his successor, telling supporters they should vote for the vice president if a new presidential election was necessary.
Bolivian President Evo Morales sent his greetings to Chavez in a New Year's Eve address.
"I'm very sorry that our Latin American brother Hugo Chavez, our ... anti-imperialist comrade, a revolutionary, is facing such a difficult situation," Morales said, wishing Chavez "a lot of strength, a lot of energy so that he can soon recover."
Morales made a quick stop in Havana last week to visit Chavez, but didn't refer to that trip.
Venezuela's government released a statement Tuesday saying that Chavez and his government congratulate Cuba and President Raul Castro on the 54th anniversary of the triumph of the Cuban revolution. The statement said that today "the circumstances of the fight for life have united more than ever before the Bolivarian and Cuban Revolutions."
Chavez's socialist-inspired Bolivarian Revolution movement has long had close ties to Cuba and Chavez's mentor Fidel Castro. Chavez has undergone much of his cancer treatment in Cuba, where he originally announced he had been diagnosed with the illness.
During Chavez's presidency, the Cuban government has aided its ally by sending thousands of doctors and nurses to Venezuela along with other specialists, providing free health care to the poor while the South American country has shipped oil to Cuba under preferential terms in return.
The Venezuelan government said in its statement that Cuba "can count on the Bolivarian Revolution to repay all the love and solidarity that it has lavished on our nation in its struggle to be every day freer and sovereign. Today we all say with Chavez, long live Fidel! Long live Raul! Long live Cuba and its revolution!"
Associated Press writers Christopher Toothaker in Caracas and Paola Flores in La Paz, Bolivia, contributed to this report.