Sophie Turner and Joe Jonas had four days of “productive mediation” last week and are close to reaching a settlement in their custody battle case, their lawyers wrote in a letter to a judge Tuesday.

The estranged celebrity couple asked a Manhattan federal court judge to pause the case and cancel the Jan. 2 trial date set during a hearing a week ago – where Turner, 27, phoned in and Jonas, 34, was not present.

“Due to the progress made at mediation, the parties believe that an amicable resolution on all issues between them is forthcoming,” attorneys for both sides wrote in a letter to the court.

The “Game of Thrones” actress on Sept. 21 filed an emergency lawsuit against the Jonas Brothers rocker, seeking for their two daughters Willa, 3, and Delphine, 1, to be returned to their “forever home” in England.

The pair attended mediation in the Big Apple from Oct. 4 through Oct. 7, after having agreed to keep their girls in New York while the case played out.

Turner and Jonas also reached a temporary custody schedule through the new year – which allows either side to travel with the children either in the US or in the UK.

Each parent will have the kids – joined by the nanny – for roughly 20-day periods before handing the girls off to the other, the court papers show.

Turner starts of the rotation with the girls from Oct. 9 through Oct. 21, when she will have to bring them back to New York for Jonas to take over.

At the hearing last week, Stephen Cullen, a lawyer for “The Dark Phoenix” actress, said that both Turner and Jonas had been “seeing their children in a very unfettered way.”

Jonas filed for divorce in a Florida court on Sept. 5.

The couple have a Dec. 14 hearing in the divorce case – which they will both be required to attend in-person.

It was not immediately clear if the impending settlement in Turner’s Manhattan lawsuit would encompass the divorce case or solely the custody case.

Turner, in her suit, claimed she and Jonas had been set to move into a stunning Wallingford, Oxford estate by December, making the UK their daughter’s home base.

It was not immediately clear if the impending settlement in Turner’s Manhattan lawsuit would encompass the divorce case or solely the custody case.

But then Jonas suddenly filed for divorce and allegedly tried to keep the girls from relocating to the UK – including by withholding their passports from Turner, her suit claimed.

Lawyers on both sides didn’t return requests for comment Tuesday.