The royal couple - who are expecting their first child in mid-July - are said to have picked Philip as one of their favourite names, according to the Daily Star.
Bookmakers are currently offering odds of 20-1 on Philip being chosen as the child's name.
Prince William's full name is William Arthur Philip Louis, while his father is Prince Charles Philip Arthur George.
A royal insider said: "The name Philip is in the frame for a boy. It will certainly have it somewhere in his name.
"It is popular with them at the moment and could even be the first name as William is very fond of his grandfather."
Kate Middleton recently revealed that she has a "shortlist" of baby names, saying that friends often suggest her ideas.
Prince William is said to be keen on having a girl, while Middleton would prefer a baby boy.
Photo gallery - the Duchess of Cambridge in pictures: