The baby girl was born on Monday and is the pair's first child.
A spokesman said: "They chose Sophia Ruby because they just loved the names."
Having left the hospital with the baby, Clancy is thought to be recovering well and the couple are said to be "thrilled".
According to The Sun, the Spurs footballer was present at the birth in a London hospital.
The pair reunited in August, following allegations that Crouch had cheated on her.