The hearing will represent the final chance for Bamber to have his whole life tariff overturned following a three-and-a-half-year human rights battle to have the UK's most severe punishment torn up.
It follows a ruling made by European judges in January that "whole life" sentences are lawful.
The three killers had argued in the European Court of Human Rights that ordering someone to spend their entire life behind bars without parole was tantamount to degrading and inhuman treatment.
The Strasbourg court dismissed the case and said all three had committed "particularly brutal and callous murders".
The judges ruled that condemning people to die in jail was not "grossly disproportionate" and that in each case, London's High Court had "decided that an all-life tariff was required, relatively recently and following a fair and detailed consideration".
That ruling will now be tested by the court's Grand Chamber after a panel of five judges granted the appeal by killer Douglas Vinter, who stabbed his wife in February 2008.
Vinter's appeal means the cases of Bamber, who killed his adoptive parents, sister and her two young children in August 1985, and Peter Moore, who killed four gay men for his sexual gratification in 1995, will also be considered by the Grand Chamber judges.
Bamber, 51, has always protested his innocence and claims his schizophrenic sister Sheila Caffell shot her family before turning the gun on herself in a remote Essex farmhouse.
In 2009, he lost a Court of Appeal challenge against the order that he must die behind bars. He has twice lost appeals against conviction.
The Criminal Cases Review Commission reached a provisional decision not to refer his case back to the Court of Appeal last February despite claims by his legal team that they had new evidence that could overturn his conviction.
His solicitor, Simon McKay, said today that he was "obviously delighted" with the latest development.
Mr McKay added: "It demonstrates that his case remains arguable and he looks forward to presenting his position at the Grand Chamber in due course.
"It is part of his long battle to challenge the home secretary of the day going beyond what the trial judge said would be the appropriate sentence he should receive."
The appeal will be heard in Strasbourg on November 28, a court spokesman said.