Chavez is “in critical condition, not grave, but critical, complicated,” the source told the newspaper; the Spanish-language, sister publication of the Miami Herald.
Officials say Chavez underwent surgery June 10 for a pelvic abscess, which is an accumulation of pus that can have various causes, including infection or surgical complications. Neither Chavez nor doctors treating him have disclosed what caused the abscess.
Nobody has heard Chavez speak since he talked by telephone with Venezuelan state television on June 12, saying he was quickly recovering from the surgery two days earlier.
Venezuelan Vice President Elias Jaua said Chavez is attending to his day-to-day government duties while recuperating.
Venezuelans are accustomed to near daily speeches and television appearances by Chavez that can last several hours, even when he's traveling abroad.
Chavez's Twitter page carried a message on Friday saluting Venezuela's military on a national holiday, though he did not provide any information about his health.
"A big hug to my soldiers and to my beloved people," the message read. "From here, I am with you in the hard work every day."
Foreign Minister Nicolas Maduro urged Venezuelans on Friday to wish for Chavez's complete recovery and express their "most authentic love so that his health is re-established."
"We've maintained constant communication with him and he's informed of all country's events," Maduro told state television.
Maduro offered no details on Chavez's health.
Authorities have also sought to quash such talk.
"In response to all the rumors, I can testify that the president is recovering in a satisfactory manner," Adan Chavez, one of the leader's brothers who is a state governor, told state television Wednesday. "The president is a strong man."
He added that "it's not clear" when his younger brother would return home, but said the president is expected to leave Cuba within 10 to 12 days.
The Associated Press contributed to this report.