The two men met at Benedict's home in the grounds of the Vatican so Francis could be the first to wish him many happy returns.
While the pope emeritus, who turns 90 on Sunday, keeps a low profile most of the time, Francis makes sure he is included in key celebrations in the Catholic Church and brings new cardinals to meet him.
In 2013, Benedict became the first head of the church to resign in seven centuries, amid speculation he was on the verge of a nervous breakdown, unable to cope at the top of an institution beset by scandals.
"He is serene, calm and in a good mood. He wants to do something small (for his birthday) to match his strength," Benedict's personal secretary Georg Gaenswein said in an interview with the Repubblica daily on Wednesday.
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"His brother Georg will come to visit for a few days, and this is the biggest present he could ask for. On Easter Monday, the day after his birthday, there will be a small party," he said.
Benedict was once a keen piano player and just a year ago was playing Mozart from memory, but now "he says his hands no longer obey him as they used to, or at least not enough to play well".
He said the elderly German had "never regretted" stepping down.
Pope Francis, 80, who was elected shortly after Benedict resigned, has said of the unusual situation of having two popes living so close to each other: "It is like having a grandfather -- a wise grandfather -- living at home.