Pope Francis marked a week in hospital with pneumonia Friday, as speculation grew over the 88-year-old's ability to continue leading the Catholic Church.

Francis, admitted to Rome's Gemelli hospital with breathing difficulties, has slightly improved over the past couple of days despite pneumonia in both lungs, according to the Vatican.

"The night went well. This morning Pope Francis got up and had breakfast," it said in a regular morning briefing Friday, adding that the next update would come earlier than usual, at around 5:30 pm (1630 GMT).

But that has not dampened questions over the future of a leader with a punishing domestic and international schedule who has been increasingly plagued by health issues in recent years.

Francis has disappeared from public view since his admittance to a special papal suite on the 10th floor of the Gemelli.

The Vatican said he is cheerful and has been alternating rest with reading, but has yet to publish the standard photographs of him in hospital that have marked his previous stays, fueling rumors.

"I know that some out there say my time has come, they are always bringing me bad luck!" Francis is reported by Italian media to have quipped to Italy's Prime Minister Giorgia Meloni when she visited him on Thursday.

The Argentine is no stranger to the plotting and scheming his health woes inevitably prompt among those keen to get their man in as next pope.

After undergoing colon surgery in 2021, he joked to a group of Jesuits that "I'm still alive. Even though some wanted me dead".

"They were preparing the conclave", he said, in reference to the meeting of cardinals to elect a new pope after a death or resignation.

'Vital function'

Francis -- who had part of one of his lungs removed as a young man, making him more prone to respiratory diseases -- has undergone hernia as well as colon surgery in the past four years.

He is overweight and suffers constant hip and knee pain, which force him to use a wheelchair most of the time.

He is also one of the oldest popes ever -- and though he has said the job is for life, Francis has also left the door open to resigning like his predecessor Benedict XVI.

Italian Cardinal Gianfranco Ravasi told the Corriere della Sera Friday he did not rule out Francis stepping down.

"Francis himself has said he had already signed a letter of resignation at the beginning of his pontificate," he said.

The fact that "a fundamental vital function such as breathing" was compromised complicates the matter, Ravasi told the paper.

"The knee is one thing, but if one feels that the entire body is in difficulty, it's another thing."

French Cardinal Jean-Marc Aveline also said that while "completely confident in the lucidity of the pope", he would not rule out a resignation.

"If he considers that it is the best thing for the good of the Church, he will do it," he told journalists on Thursday.

Rumors the pope was far sicker than believed, however, were "fake news", Ravi said.

And whether or not he is contemplating retirement, Francis's "great desire is to at least complete the Jubilee", which began in December and is a year of Catholic celebrations, Ravasi said.

"He feels it is his great moment", he said.