Australian mother Sally Faulkner has asked estranged husband Ali Elamine to drop Lebanese abduction charges against her in exchange for her renouncing all claims to custody and cooperating in getting a divorce.

Key points: Faulkner to give up sole custody granted by Family Court in Australia if husband drops charges

Faulkner to give up sole custody granted by Family Court in Australia if husband drops charges Ruling allowed Australian police to get children back but Faulkner did not register it in Lebanon

Ruling allowed Australian police to get children back but Faulkner did not register it in Lebanon Team paid by Faulkner, who used money given to her by 60 Minutes, her lawyer says

Ms Faulkner and four members of a Channel Nine 60 Minutes crew were among a group of people arrested after an attempt to take Ms Faulkner's two children off the streets of the capital, Beirut.

If Mr Elamine agrees to drop the charges, Ms Faulkner will give up sole custody granted to her by the Family Court in Australia.

The Australian Family Court ruling, granted on December 15, even allowed Australian police or agents appointed by Ms Faulkner to get her children back — but she did not register it in Lebanon.

Ms Faulkner's Lebanese lawyer Ghassan Moghabghab said: "It's a very strong judgement and we are sorry that it's not being used .

"Especially when you read the articles of the judgement, it was obviously 100 per cent to her favour."

Mr Elamine got his own custody ruling from a religious court in Lebanon, but it is not clear when it was issued.

The ABC has been told the Lebanese judge does not view the recovery as a kidnapping, but rather as a mother trying to reunite with her children.

Ms Faulkner hopes for the right to see her children whenever she wants in Lebanon, Australia or a third country.

If Mr Elamine agrees to drop charges against Ms Faulkner, she would likely be released on bail and it could reduce the severity of charges against all involved.

Mr Elamine got his own custody ruling from a religious court in Lebanon. ( Supplied: Nine )

Faulkner paid for recovery with 60 Minutes' money: lawyer

The child recovery team were paid directly by Ms Faulkner, who used money given to her by 60 Minutes, Mr Moghabghab said.

But Mr Moghabghab said "I don't have any idea" if 60 Minutes paid Ms Faulkner intending for it to go towards the operation, or just for her story.

Mr Moghabghab said Ms Faulkner told a judge she regrets paying for the attempt. ( Supplied: Nine )

The payment can not be confirmed independently and the claims have not been tested in court.

Mr Moghabghab said Ms Faulkner told a judge she regrets paying for the attempt.

Earlier, authorities said they had a signed statement from a member of the recovery team saying Nine paid $115,000 for the operation.

However, that statement is uncorroborated, the ABC has not seen it and cannot confirm the claim.

Mr Moghabghab said Ms Faulkner was in the car when alleged operatives of Child Abduction Recovery International grabbed her son and daughter off a Beirut street and bundled them into the waiting car.

"Maybe she was showing the persons that were executing [the abduction] who are the children, maybe they don't know," he said.

"Maybe she was present that if the children were taken from the grandmother, they began crying, so the presence of the mother, it will be easier for them to take the children and give them to their mother."

The team made a clean getaway, but was caught not long after the fact. Soon after, the boy and girl were returned to the father and Ms Faulkner and the 60 Minutes crew were arrested.