They claim to have evidence Abdelbaset Ali al-Megrahi was pressured by ministers to drop his earlier appeal.
He was allowed to return home to Libya in 2009, suffering from terminal prostate cancer, and died three years later.
The Scottish Criminal Cases Review Commission (SCCRC) has been asked to look again at his conviction.
The involvement of Megrahi's family may overcome a legal hurdle against a bid for a re-examination of the case.
The Libyan secret service agent, the only man convicted of Britain's worst act of terrorism, was jailed for life following his conviction in 2001, and ordered to serve a minimum of 27 years.
He died in May of 2012, three years after being allowed out of prison on compassionate grounds.
Shortly before that, he had abandoned an appeal against his conviction which had been brought after a four-year investigation by the SCCRC.
Ministers have always claimed they played no part in this.
But Aamer Anwar, the lawyer who is taking the case to the commission, said: "To date both the British government and Scottish government have claimed that they played no role in pressuring Mr  Megrahi into dropping his appeal as a condition of his immediate release.
"However the evidence submitted to the Commission today claims that this is fundamentally untrue."
Mr Anwar added: "The fundamental question for the commission is whether it regards it as in the interests of justice to refer a case back to the High Court where the convicted person himself had commenced an appeal on a SCCRC reference and then chosen to abandon it?
"The answer might depend on the precise circumstances in which the appellant came to abandon his appeal.
"Mr Megrahi's terminal illness; the fact that prisoner transfer was not open while the appeal was ongoing; and whether Mr Megrahi had no way of knowing that (Justice Secretary) Kenny MacAskill would ultimately opt for compassionate release rather than prisoner transfer or, as is alleged, that he was led to believe that he would not be released unless he dropped his appeal."
Mr Anwar, who is acting for six relatives of Megrahi, who are not being named because of the volatile political situation in Libya, has asked the commission to consider the grounds of appeal before the court when the last proceedings were abandoned.
He has also asked the SCCRC to consider further evidence.
This includes claims that it was impossible for the bomb timer identified by prosecutors at Megrahi's trial to have been responsible for the bombing; and that there is evidence the bomb - which the court heard had been put onto a flight at Malta - had already been at Heathrow Airport, the flight's departure airport.
Relatives of some of the 270 victims of the bombing - led by Dr Jim Swire, whose daughter Flora died on Pan-Am Flight 103 - previously said they would seek to have the conviction reviewed.
In a statement released last month, before it was known that Megrahi's family were to making a request to the SCCRC, the commission said it would have to decide whether Dr Swire and other bereaved relatives had a "legitimate interest" to be allowed to apply for a review.
In a statement, it added: "If it is decided that Dr Swire has a 'legitimate interest' in this matter, the commission will also require to address whether it is 'in the interests of justice'."
It said it would have to consider the fact that Megrahi abandoned his appeal in 2009 and that neither he nor any member of his family had lodged an application for a further review of his conviction prior to his death.
Dr Swire said: "If the verdict is unjustified then the extraordinary delay experienced thus far in redressing it has protected the real perpetrators from the probing of international justice.
"It has also protected the systems which failed to protect our families. The review of the evidence and verdict still lie with us in Scotland.
"We still have the option of re-examining this case in Scotland. If we do not do so, it will pass to others to examine and Scotland, her people and her law would be the losers."
When the SCCRC announced in 2007 it was referring Megrahi's case back to the appeal court, its full statement of reasons extended to more than 800 pages and was accompanied by a further 13 volumes of appendices.
It is likely to be several months before the commission decides whether it can accept the case a second time.
If it does, board members will be given time to study the terms of the application and the basis for the previous review and referral.
For the last review the commission took on extra legal staff to deal with the volume of material it was given.
However, it has only retained the material which was necessary to support the case it made for an appeal hearing. The rest has been returned to legal teams who supplied it.
A Scottish government spokesperson said it was a matter for Megrahi's family or relatives of the Lockerbie bombing victims to decide whether to apply to the SCCRC to examine the case.
"It is appropriate for the SCCRC to take forward consideration of an application on that basis," the spokesperson added.
"The Lockerbie case remains a live investigation, and Scotland's criminal justice authorities have made clear that they will rigorously pursue any new lines of inquiry.
"The justice secretary is on record as saying that at no time did he or any other member of the Scottish government suggest to anyone connected with the Libyan government, or to Mr al-Megrahi himself that abandoning his appeal against conviction would in any way aid or affect his application for compassionate release.
"We have no interest in Mr al-Megrahi's appeal being abandoned and the justice secretary had no involvement in Mr al-Megrahi's decision to drop his appeal against conviction - that was entirely a matter for him and his legal team."
A Crown Office spokesman said they "do not fear scrutiny of the conviction by the SCCRC."
The spokesman added: "The evidence upon which the conviction was based was rigorously scrutinised by the trial court and two appeal courts after which Megrahi stands convicted of the terrorist murder of 270 people.
"We will rigorously defend this conviction when called upon to do so.
"In the meantime, we will continue the investigation with US and Scottish police and law enforcement."
Frank Duggan, who is president of the Victims Of Pan AM Flight 103 group in the United States, said the press reported stories "every time Megrahi's supporters take a breath".
"This trial was the largest and longest trial in history under Scottish law," he said.
"It has been appealed twice.
"There is nothing new - there are no new facts but there are people who simply will not let the case go away."
The family of the Lockerbie bomber has instructed a Scottish lawyer to apply to have his conviction reviewed.