Francis, 88, was admitted to Rome's Gemelli hospital on February 14 with what was originally described as a bronchitis and was later revealed to have developed into something more serious.
In a medical update on Thursday, the Vatican said the pontiff's condition had "continued to show improvement" but his prognosis remained guarded due to a complex clinical picture.
A Vatican official, who did not wish to be named because he was not authorised to discuss the pope's health, noted that Thursday's statement was the second consecutive one that did not describe the pope's condition as "critical".
"Maybe we can say he has passed the most critical phase," said the official.
Cardinal Michael Czerny, head of the Vatican's human development office, said in an interview with Italy's La Stampa newspaper that Francis was getting better, albeit "slower than what we would like".
Francis, who has been pope since 2013, has suffered several bouts of ill health over the past two years. He is prone to lung infections because he developed pleurisy as a young adult and had part of one lung removed.
