Menino, who will turn 70 next month, has been mayor for nearly 20 years and has not yet announced if he will run for a sixth term, the Boston Herald reported Wednesday.
Menino has been in the hospital for three weeks since returning early from Italy, and his doctors on Tuesday were vague about his condition.
A virus has "lifted," his doctor said, and a blood clot has been fixed.
The mayor has been fitted with a back brace for a fractured vertebra, but his pain level has increased in the past few days.
His doctors have not set any timetable for his release.
Still, don't count out the mayor to run for a sixth term next year, an aide says.
"No one should underestimate him ever," Menino spokeswoman Dot Joyce said. "The guy just loves his job."
Democratic strategist Michael Goldman said "the only people talking about him not being a candidate are those who want to be a candidate."
Hopefuls seen as likely to jump into any potential race are seven city councilors, a state representative, the current and former Suffolk County attorney, the Suffolk County sheriff and the state's top transportation official.