A High Court judge says he hopes that Madonna and her ex-husband Guy Ritchie can settle a dispute relating to the future of their 15-year-old son Rocco.
Mr Justice MacDonald said it was his "fervent hope" that the singer and film producer could have discussions and reach an "amicable resolution".
The judge is analysing the dispute at private hearings in the Family Division of the High Court in London.
Similar hearings have taken place in New York following the launch of family court litigation in the United States.
Mr Justice MacDonald considered a number of legal issues at a hearing on Thursday.
Another hearing is scheduled for later this month.
Mr Ritchie, a film producer, was at Thursday's hearing - but neither Madonna nor Rocco was in court.
A judge in New York has heard that Rocco is living with his father in England.
Lawyers told Mr Justice MacDonald that Madonna wanted the teenager to return to live with her in the US and had issued an application in England under an international legal convention.
They said the singer now wanted to withdraw that application and bring the legal proceedings launched in England to a close.
The judge said he had to decide whether the proceedings launched by Madonna in England should end - or whether he should make decisions on Rocco's welfare.
"It would be my fervent hope that discussions between the parties can reach an amicable resolution," Mr Justice MacDonald told the court on Thursday.
"An amicable negotiated resolution is definitely the way forward and in Rocco's best interests."
Specialist legal teams representing Madonna, Mr Ritchie and Rocco appeared before Mr Justice MacDonald.
Madonna was represented by barristers David Williams QC and Jacqueline Renton; Mr Ritchie was represented by barristers Alex Verdan QC and Michael Gration; Rocco was represented by barristers Henry Setright QC and Edward Devereux via a court-appointed guardian.
Mr Williams said Madonna had launched proceedings in England under the The Hague Convention on the Civil Aspects of International Child Abduction.
A judge in New York is analysing issues relating to the custody of Rocco - and those proceedings have already been widely reported in the media.
She has also urged Madonna and Mr Ritchie to reach agreement.
Mr Justice MacDonald heard argument from lawyers at a private hearing.
He allowed reporters to attend and allowed some information aired at the hearing to be reported.