But the family sees the move as nothing more than a further delay in the judicial process.
Miller was killed in the Gaza Strip in 2003 as he worked on a documentary about the effect of the Palestinian-Israeli conflict on people on the ground.
An inquest into his death held in Britain last year found that he was unlawfully killed by a single shot fired by an Israeli soldier.
In 2005, the Israeli army had said it would take no action against an officer accused of involvement in the shooting, citing lack of evidence.
A spokesman for the British attorney-general's office said on Tuesday: "We have received a letter from the Israeli attorney-general ... it asks for further information and we're making arrangements to take that forward."
Video analysis
Earlier this year, Lord Peter Goldsmith, Britain's attorney-general at the time, wrote to Meni Mazuz, his Israeli counterpart, urging him to consider the new analysis of the videotape.
He gave the Israeli government six weeks to reply to a request to reopen the case.
The Israeli army has claimed that Miller was hit in crossfire as he left a building in a refugee camp in Rafah, but analysis of the sound of the gunfire on the videotape by British police concluded otherwise.
He is shown on the video wearing a helmet and a bullet-proof vest with "TV" written on before he steps into darkness and the first shot is heard.
The group can then be heard shouting that they are British journalists before two more shots are fired.
Anne Waddington, Miller's sister, told Al Jazeera that she was unimpressed with Israel's response to British calls to reopen the inquiry.
"The information that they are requesting they have actually had for four and a half years now and the only difference is that their evaluation was commensurate with their intention to evade any responsibility," she told Al Jazeera.
"We would ask the British government to continue to assert pressure on the Israelis until we actually have some form of constructive response not a further delay."