Madonna’s lawyer told judge Alistair MacDonald that she wanted to end the court battle being fought in London with British film director Guy Ritchie over their 15-year-old son.
The former couple are also engaged in similar court action in New York.
MacDonald finished hearing evidence on Friday and now has to decide whether the proceedings in the English High Court should close, or if he should make decisions about Rocco's welfare.
A New York judge in December asked Ritchie to send Rocco back to the US, after the teenager decided to move to London rather than stay in his mother's home or accompany her on tour.
Ritchie has attended the London court hearings, though Madonna has been singing in Australia and New Zealand.
"What she has always wanted to do... is to find a way in which this family can get to heal the wounds which have been inflicted on this family over the past four months," said Madonna's lawyer David Williams.
The singer wants to "chart a course for Rocco and the family which enables them to put this behind them and to restore peace to the family," he said, adding that Madonna wants them to meet in New York over Easter.
Ritchie's lawyer, Alex Verdan, said the director had proposed a meeting in London as the pair had not been in the "same place at the same time" since the dispute began.
Earlier this week, judge MacDonald said little detail could be revealed about the hearings, but relaxed his restrictions following an application from two British newspaper publishers.
Madonna and Ritchie split in 2008 after eight years of marriage.
The pop icon touched on the custody battle in a concert Sunday in New Zealand, where she dedicated Edith Piaf's signature song "La Vie en Rose" to the 15-year-old.
"There is no love stronger than a mother for her son," she said tearfully before singing the song in Auckland.
"I hope he hears this somewhere and knows how much I miss him."