"We will see how he reacts to treatment," Vatican press office director Matteo Bruni said on Saturday.
"I don't have a precise date (for release)."
The Pope slept well in hospital on Friday night, and his fever did not rise overnight, Bruni said.
Francis was expected to continue receiving diagnostic tests on Saturday, he said.
A Vatican official, speaking on background without authorisation, said the Pope did not have a fever on Saturday morning.
The pontiff, 88, has been suffering from bronchitis for more than a week and was taken to Rome's Gemelli hospital for treatment on Friday morning.
Gemelli hospital, Rome's largest, has a special suite for treating popes, and is known especially for treating the late Pope John Paul II often during his long papacy.
Francis spent nine days at Gemelli in June 2023, when he had surgery to repair an abdominal hernia.
Outside the hospital on Saturday, groups of people were gathering under a famous statue of John Paul II to pray for Francis.
"We found out yesterday morning, with sadness," said Giovanni Di Muro, an Italian who was visiting his son in hospital.
"We hope it's nothing serious and that everything will be fine."
The Vatican said on Friday evening that diagnostic tests indicated Francis had a respiratory tract infection.
Francis, who has been pontiff since 2013, has had influenza and other health problems several times in the past two years.
As a young adult, he developed a case of pleurisy and had part of one of his lungs removed, and has been prone to lung infections in recent times.
Francis has been suffering from respiratory issues since mid-December.
He begged off reading statements at several of his public events in January and February, attending the occasions but asking aides to read his prepared remarks.
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