Mel Gibson will be served with legal papers this week asking him to admit he is the father of a 29-year-old woman resulting from a one-night stand in the back of a car in Australia in 1976.
A report in the News of the World says Gibson, 50, will be required to undergo a DNA test if he denies paternity of Carmel Sloane.
The tabloid newspaper did not say where Carmel is from.
"I'd love it if he recognised us as family," said Carmel, a painter who has a 10-year-old son.
"I'm not looking for a meal ticket, I'm happy with my life.
"I just want to meet the man I've always known was my dad and for him to get to know his grandson.
"I have grown up being told Mel Gibson is my father. I've been told it for 20 years.
"I would like some kind of closure either way. I don't want it to go on for another 20 years."
Gibson, who has recently been the subject of controversy for anti-Semitic remarks he made during a drunken arrest in the US, is the father of seven children with his wife Robyn.
Carmel's mother Marilyn said she met Gibson in Adelaide in 1976 when he picked her up while she was hitch-hiking to Sydney.
She said Gibson promised to drive her there, but first ducked back to his house and came back to the car with a mattress and pillows and put them in the back of his station wagon.
"We got on really well. He was charming and funny. He said he was going to be a famous movie star," Marilyn said.
"He said he was going to change his name, but I told him not to. I said Mel Gibson was a really nice name and it suited him.
"When we stopped for the night, Mel got in the back on the mattress but I stayed in the front.
"Eventually he persuaded me to join him in the back. I told him 'if anything happens and I get pregnant I'll come looking for you'. He replied 'I'm going to be famous, you'll always know where to find me'.
"We spent the night making love and talking, but when it got light Mel said he had to be back at work at an orange juice factory in Adelaide.
"I got out of the car and never saw him again. It broke my heart to walk away."
Marilyn gave birth to Carmel in January 1977, but did not think of Gibson again for eight years until she saw him in a poster for Mad Max 3.
She then started keeping a scrapbook of the movie star that she intended to give to her daughter later in life.
"I would never have told Carmel who her father was but my mother blurted it out one day," Marilyn said.
She said she twice tried to contact Gibson when he was back in Australia, but failed to speak to him and is now hoping the legal avenue will bear fruit.
Carmel said while she would like to meet Gibson she "wouldn't expect to be part of his life".
"I'm not looking for a piece of the pie or that lifestyle."
The News of the World said it put the claims to Gibson's representatives, but they failed to respond.
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