The 88-year-old pontiff is being treated at Rome's Gemelli hospital, where he was admitted on February 14 after struggling with breathing difficulties for several days.
Double pneumonia is a serious infection that can inflame and scar both lungs, making it difficult to breathe.
On Thursday, the Vatican said for the second day in a row that Francis' condition was slightly improving, adding that "hemodynamic parameters continue to be stable."
Hemodynamic parameters refer to the body's ability to regulate the flow of blood through its organs and tissues.
The Vatican had said previously that Pope had a polymicrobial infection, which occurs when two or more micro-organisms are involved, adding that he would stay in hospital as long as necessary to tackle a "complex clinical situation".
A Vatican official, who did not wish to be named because he was not authorised to speak about the Pope's condition, said Francis was not on a ventilator and was breathing on his own.
On Thursday, the Pope was able to move around his hospital room, was taking some phone calls and was continuing to do some paperwork, the official said.
Two senior Catholic prelates expressed hope for the Pope's recovery.
"I think there is a feeling of hope, and he will come through it, thank God," said Spanish Cardinal Juan Jose Omella.
Italian Archbishop Giuseppe Satriano called the Pope a "fighter" and said: "He will win this battle."
A wave of messages of support for Francis came in from across the world, the Vatican's official media outlet reported. Several national bishops' conferences were also holding special days of prayer for the Pope.
Rome resident Adele Cucinotta was outside Gemelli Hospital on Thursday and said she was grateful to hear his condition had improved. "I am very happy that he is a little better," she said.
Francis received a visit on Wednesday from Italian Prime Minister Giorgia Meloni, his first known VIP visitor in hospital. She said he was "alert and responsive".
The Pope joked with Meloni about some people betting on his death, Italian newspaper Corriere della Sera reported on Thursday.Â
"He hasn't lost his proverbial sense of humour," Meloni said in a statement.
The pontiff's latest illness is the latest in a long history of health problems he has suffered over the years.
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