The Sun reports that the sports agent and the actress asked Becks soon after little Sailor was born in February.
An insider told the paper: "Becks and Dave are close pals and go back years. They speak to each other all the time and their families are well connected.
"They were best men for each other's weddings so are involved in their major events.
"Dave knows Becks will always be a great influence for his kids. He's the perfect choice (for godfather)."
Dave and Liv, who is the daughter of Aerosmith rocker Steven Tyler, were introduced by their mutual friend Kate Moss.
London-born Dave has Grey, seven, from his marriage to former Hollyoaks star and ex-wife Davinia Taylor, while Lord of the Rings elf Liv has Milo, eight, with ex Royston Langdon.
Dave is already a godfather to Becks' eldest son Brooklyn, a role he shares with Elton John.