Hungary's justice minister Tibor Navracsics has witten to his Irish counterpart Alan Shatter to express his displeasure at Ireland's unwillingness to hand over Ciarán Tobin.
The Dublin man was found guilty in absentia for killing two-year-old Petra Zoltai and her five-year-old brother Marton in a road accident in Hungary 12 years ago.
Mr Navracsics also sent a letter to the EU Justice Commissioner Viviane Reding and made the letters available to state news agency MTI.
Both letters call for the extradition of Tobin to Hungary, but the Supreme Court rejected Hungary's initial request in 2004 and again this month, arguing that Tobin had left the country legally before his sentence was passed.
Tobin was handed a prison term in 2002, two years after his car mounted a pavement and hit the children. He had been allowed to leave Hungary on bail before his sentencing.
The Department of Justice and Equality has told Mr Navracsics that no further action will be taken in the case.
Hungarian newspaper Magyar Nemzet has reported that children's father, Bence Zoltai, will return to Dublin in the next few days to continue his fight for justice.
In his letter to Mr Shatter, Mr Navracsics said the case has received great public attention in Hungary and it is regrettable that the extradition proceedings were terminated without the handing over of Tobin.
In November 2011, Tobin offered to enter an Irish prison on the basis that any time spent there should be deducted from any time he might have to spend in a Hungarian prison.
He was put in custody but has since been released.
The case was recently back in the limelight when Tobin won another appeal in the Supreme Court to prevent his extradition.
The Department said it cannot confirm that it has received the letter, but it is aware of the concern of the Hungarian authorities about the case.
The solicitor representing Tobin said because the case was ongoing it would be inappropriate to comment.
Asked why it was still ongoing after the Supreme Court had ruled, Catherine Almond of Garrett Sheehan and Partners said that while the Supreme Court had decided the case and that its decision was final, the case remained within the remit of the Court as it was being absolutely finalised.
As a result, she said, she could not comment on the remarks of the Hungarian justice minister.