Australia's newest celebrity baby, Sunday Rose, looks like her father Keith Urban and has a "bit of a reddish tint" to her hair, her mother Nicole Kidman says.

Kidman and Urban returned to Australia on Wednesday night with Sunday Rose for their first trip back since the birth of their daughter in the US on July 7.

The couple, who were greeted by a small group of media when they flew into Sydney on a private plane, called a Sydney radio station on Thursday to plead with journalists and photographers to back off while they show their new daughter their home town.

The proud parents enthused about their love for their "little four-weeker".

"She looks like Keith. I think she does," Kidman told the Kyle and Jackie O breakfast radio show on 2Day FM.

"She's got a little bit of hair, it's a bit of a reddish tint."

Urban said being a new dad was an "awesome" feeling that only other dads could understand.

"Anybody out there who's got kids (knows) it's just a certain kind of feeling that you can only experience when you do it," he said.

"She's just awesome ... cute little thing.

"I'm not looking forward to having to go on tour and having to leave."

Kidman said she had enjoyed a "surprisingly" smooth labour, with her husband, mother and sister by her side and she was now looking forward to introducing her little "doll" to the rest of her family.

Kidman said her labour had been made easier by the presence of her mother, who is a nurse and midwife, and her sister Antonia who has had four children.

"They say a lot of it (labour) has to do with how your sister and your mother gave birth, so luckily my sister's had four children ... and that was pretty easy for her too," she said.

"My mum was a nurse too and also she's a midwife so we were surrounded ... but we also had wonderful nurses at the hospital.

"I'm embarrassed now. That's too much information on a breakfast show."

The couple pleaded with photographers to let them introduce Sunday Rose to Sydney and her other three grandparents in peace.

"Keith and I are both appealing to the press and stuff just to give us a little space so we can walk around Sydney and show the baby our town," Kidman said.

"She's tiny. She's like a doll, she's like a little, little thing.

"Just (don't photograph) right in her face or in our faces because it's scary for her."

Urban said he understood the media interest, but that people had to think about the effect it could have on their little girl.

"I get it. I get the interest there is," he said.

"But at the same time it's our little girl.

"Sometimes when people come right up in your face and you think: 'Good God, would you do that to anybody else's child?' That's all."

The media followed the family to their exclusive Darling Point mansion with the couple using several decoy cars to try to avoid photographers.

Sydney paparazzo Jamie Fawcett, who has a history of tense encounters with the 41-year-old Australian actor, was pulled over by police while pursuing the couple to their home, a Network Ten news report said.

Fawcett was allowed to leave a short time later.

No photographs of Sunday have been released and Urban confirmed the couple had no intention of selling any.