The singer appeared in court in the Malawian capital Lilongwe to make the application.
Madonna previously adopted two children from Malawi: David Banda in 2006 and Mercy James in 2009.
The 58-year-old star has two other children – Lourdes and Rocco – from previous relationships. Last year saw her involved in a custody dispute over Rocco with ex-husband Guy Ritchie.
Madonna eventually lost the case, with 16-year-old Rocco moving to London to live with his father.
She has yet to comment on her attempts to adopt two further children, but a government spokesman confirmed she had appeared in High Court on January 25.
Madonna flew to Malawi on January 24, fresh from the Women’s March in Washington.
Court spokesman Mlenga Mvula said the star would have to wait a week before hearing the court’s decision.
In 2013, Madonna explained her decision to become an adoptive parent in an article for Harper’s Bazaar: “I decided that I had an embarrassment of riches and that there were too many children in the world without parents or families to love them.
“I applied to an international adoption agency and went through all the bureaucracy, testing, and waiting that everyone else goes through when they adopt.”
However, Madonna was vilified in the press with some – including relatives of the adopted children – questioning her suitability as a parent.
The star called the furor “a real low point in my life”.
Madonna’s relationship with Malawi dates back to 2006, when she established the Raising Malawi foundation, with the goal of improving children’s lives.
She initially planned to build a $15 million girls’ academy, but later changed strategy, and used the money to fund a number of schools.
Raising Malawi also provides scholarships to female students, and is currently building the country’s first pediatric intensive care unit at the Queen Elizabeth Central Hospital, in Blantyre.
Madonna visited the project last summer, and took David and Mercy to visit the orphanages where they lived before being adopted.
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