Pope Francis' condition is improving but he still requires various in-hospital therapies, the Vatican said Saturday.

The 88-year-old pontiff marked one month in Rome's Gemelli hospital on Friday, where doctors for days have said his condition is stable and slowly improving after a critical period marked by breathing crises that risked his life.

But on Saturday, the Vatican cut short any speculation he could be imminently released, cautioning that, despite the progress, he still required therapies to be administered within the hospital environment.

"The clinical conditions of the Holy Father have remained stable, confirming the progress shown in the last week," the Vatican said in an evening bulletin on his health.

"The Holy Father still requires hospital medical therapy, motor and respiratory physiotherapy; these therapies, at present, are showing further, gradual improvements," it added.

It said he was continuing to receive "high-flow oxygen therapy", which gradually lowers "the need for non-invasive mechanical ventilation during the day".

The pope is still receiving "thousands" of letters a day in support from around the world, Italy's postal service said.

Well-wishers have been flooding the city's postal service with letters "from the time of his hospitalisation until today," the postal service said in a statement.

The "surge" in correspondence was being felt at the Fiumicino postal sorting centre, with "up to 150 kilograms more mail per day," said the unit's head, Antonello Chidichimo.

"These days, the flow of letters and messages addressed to Pope Francis is particularly intense, a sign of the affection and closeness of the international community," said the statement.

In a bit of self-promotion, the postal service said it was playing a "fundamental role in ensuring that every message of support and prayer reaches its destination".

The Vatican has yet to say when Francis could be released from the hospital, where he is in a special papal suite on the 10th floor.

On the anniversary of Francis's 12 years as pope Friday, his favourite football team -- Buenos Aires' San Lorenzo club -- posted a video message of support, with other messages arriving from Brazilian footballer Neymar and Italy's former player and manager Ciro Ferrara.

The Argentine pontiff has previously thanked well-wishers for their letters and prayers, noting their "affection and closeness".

In February, ten days after Francis was hospitalised, the Vatican published a message written by him in which he cited the "many messages of affection".

"I have been particularly struck by the letters and drawings from children," wrote Francis.