The Pope was admitted to Rome's Gemelli hospital on February 14 after experiencing difficulty breathing for several days and subsequently had pneumonia diagnosed in both lungs.
The Vatican described his condition as critical for the first time on Saturday, reporting that the 88-year-old Francis had needed supplemental oxygen and blood transfusions that day after a "prolonged asthma-like respiratory crisis".
"The night was tranquil. The Pope rested," the Vatican said in an update on Sunday morning without providing further information.
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 awake and alert on Sunday.
The Pope was receiving oxygen as needed via a small tube under his nose but was breathing unaided, the official said.
A further update on the Pope's condition is expected on Sunday evening after clinical exams during the day.
In a written message for his usual Sunday prayer in St Peter's Square, which the Pope was unable to give for a second consecutive week, Francis said he was continuing "confidently" with his treatment in hospital.
He thanked his doctors and people who have sent him messages of support.
Archbishop Rino Fisichella, a senior Vatican official, told participants at a mass in St Peter's Basilica on Sunday morning they should make their prayers for Francis "stronger and more intense".
The Diocese of Rome, which the Pope leads, announced a special mass for Sunday evening to pray for Francis so he would have "the strength necessary to get through this moment of trial".
Double pneumonia is a serious infection that can inflame and scar both lungs, making it difficult to breathe.
The Vatican has described the Pope's infection as "complex", saying it is being caused by two or more microorganisms.
Francis, who has been Pope since 2013, has suffered bouts of ill health in the past two years.
He is particularly prone to lung infections because he developed pleurisy as a young adult and had part of one lung removed.
Near the Vatican on Sunday morning, pilgrims were expressing concern for the Pope's condition.
"I am very, very sad," said Elvira Romana, from Italy.
"I don't know how you can continue on normally at this moment."
Outside Gemelli hospital, groups of people were gathering to pray near a statue of the late Pope John Paul II, who was treated at the facility many times during his long papacy.
People were leaving flowers and notes for Francis, and lighting candles at the base of the late Pope's memorial.
In a statement on Saturday evening, the Vatican said the Pope's respiratory crisis had required the administration of "high-flow oxygen".
It said he had also needed blood transfusions because tests showed he had a low platelet count, which is associated with anaemia.
Platelets are cell fragments in the blood that form clots and stop or prevent bleeding.
In a briefing on Friday, two of his doctors said the Pope was highly vulnerable because of his age and general frailty.
Dr Sergio Alfieri, a senior member of the Gemelli staff, said there was a risk the lung infection could spread to his bloodstream and develop into sepsis, which "could be very difficult to overcome".
Doctors said the Pope was highly vulnerable due to his age and frailty.
