Pope Francis told a meeting of the heads of Vatican offices last week that he was moving against Burke because he was a source of “disunity” in the church.
The pope is recovering from the flu and lung inflammation, the Vatican said.
Pope Francis held separate meetings at the Vatican with relatives of Israeli hostages and Palestinians living in Gaza.
Bishop Joseph Strickland had said in interviews he would not leave his post voluntarily.
Pope Francis responded to questions from five conservative cardinals in a way that marks a reversal from the Vatican's current official position.
“Those who risk their lives at sea do not invade, they look for welcome, for life," Francis told French President Emmanuel Macron.
Pope Francis, who embraces tackling issues like climate change and poverty, said a correct understanding of Catholic doctrine allows for change over time.
The 80-page book, published in 1995 but no longer in print, is being used to blast Pope Francis' appointment of Archbishop Victor Manuel Fernández.
The pope is expected to remain at Rome’s Gemelli hospital for several days and all papal audiences through June 18 have been canceled as a precaution.
In July 2021, Francis spent 10 days in hospital to remove 13 inches of his large intestine after suffering severe inflammation and narrowing of the colon.