Nguyen Phu Trong, who is both president and head of the ruling Communist Party, vanished from the public eye after falling ill on 14 April.
Speculation intensified on Friday when the 75-year-old was absent from the funeral of former president Le Duc Anh.
State media have not explained why Mr Trong, who was due to preside over the state funeral, did not appear.
A government spokesperson said prior to the funeral that the president’s health had been affected by “the high intensity of his workload and weather”.
The spokesman added that Mr Trong would return to work soon, but his failure to appear at the state funeral has fuelled speculation.
Discretion over the health of party leaders and government officials is a way to portray Vietnam as a stable nation under single-party rule, said Giang Nguyen, news editor of the BBC’s Vietnamese service.
“Last November, Vietnam passed a law to classify the health of top party leaders and government officials as ‘state secrets’, in an attempt to protect them from malicious rumours and what they consider ‘anti-government attacks’ by hostile forces,” he said.
“This piece of legislation, although it has not yet come into life, already made local journalists very cautious when reporting about the alleged illness of President Nguyen Phu Trong, leaving room to social media and foreign media to speculate about his situation.”
Despite his age and image of a conservative ideologue, Mr Trong has gained popularity thanks to an anti-corruption campaign which led to the arrest of former ministers and police officers.
Mr Trong met North Korean leader Kim Jong Un and US president Donald Trump in February when talks were taking place between the two countries in Vietnam.