Long-term girlfriend Emma Rhys-Jones gave birth to a baby girl in October and it is believed Bale is happy to delay a move to provide his family with some stability.
Mabbutt told the Daily Star Sunday: "Why are people talking about Bale going abroad?
"He's playing the football of his life, he's got a very close family.
"His family travel to see him play every week.
"And he's on a very good four-year contract he signed at the start of the season. Hopefully we can get in the Champions League - what more could he want?"