The new parents chose the moniker more than a week after their first child arrived on March 22.
The Sun said the baby has his father's surname.
A source told the newspaper: "Cheryl and Liam wanted to get to know their baby before deciding on a name.
"They spent over a week getting to know him first before deciding so they got to know him first.
"They didn't have the name Bear before the birth.
"But they'd been thinking of a name since meeting him and they just felt it suited him."
The source said the couple are "truly over the moon and loving being parents for the first time together".
Cheryl, 33, and Payne, 23, have been keeping a low profile since their first child was born.
However, shortly after he arrived they both shared a sweet image on social media showing the One Direction star cradling their baby.
Actresses Kate Winslet and Alicia Silverstone both have sons named Bear.