According to Sofia Coppola, writer/director of the movie, in this Vanity Fair article:
I’m pretty sure it was our last day of shooting. We were exhausted, so I think it helped that they really were saying goodbye. I didn’t intend [the end] to be a big thing. He whispered something between them. It came from the tradition of Italian movies—they would just say numbers and figure out the dialogue later. [But] then we left it. We never seemed to be able to sum it up. I always liked that Bill Murray says it’s between the two of them. [Everybody] asks what he whispers to her. I just don’t get why it’s such a thing, but I’m touched that people feel connected to it.
She offered a similar explanation recently:
Even today, she’s coy about the content of the intimate whisper, saying it’s open to interpretation.
“I was thinking about the Italians, because they used to film and add the sound later,” Sofia explained on the fourth episode of BBC Arts’ Life Cinematic, an series of in-depth explorations of film for BBC Four. “I didn’t intend for it to be silent and then in the editing we were like, ‘Oh it’s better if it’s just between them and the audience puts their own interpretation,’ it’s so much stronger that way.”
That episode is available here (after it is broadcast on 25 November 2020).