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Lawyers for the 57-year-old, who is dying of prostate cancer, said he had taken the unexpected decision after his condition took a "significant turn for the worse".
He is expected to be freed next week on compassionate grounds, but relatives of the victims had expected his appeal to continue, even after his death.
They reacted to the latest news with dismay and immediately renewed their calls for a full public inquiry into the atrocity in which 270 people died.
The decision means that Megrahi, who has always protested his innocence, will die a guilty man.
He had previously claimed that he would not return to Libya until he cleared his name, and politicians expressed concern that he may have been put under pressure as part of a deal to return home.
Scottish ministers are thought to be preparing to free him after being persuaded that he does not have long to live.
He lost his original appeal in 2002 but was granted a second appeal after an independent review body ruled that he may be the victim of a miscarriage of justice.
The Scottish Executive, which insists no final decision has been taken on the Libyan's fate, said it was not involved in the appeal decision.
Reports that Kenny MacAskill, the Scottish justice minister, has already taken the decision to free him next week have infuriated American relatives of the Lockerbie victims.
They are convinced of his guilt and believe he should die in jail in Scotland, where he has served only eight years of a minimum sentence of 27 years. Most of the British families believe he is innocent and support his release.
Lawyers confirmed that he had asked for the appeal to be dropped and the request will be heard in the High Court in Edinburgh on Tuesday.
Pamela Dix, whose 35-year-old brother Peter died in the bombing, said she was disappointed by the decision.
Speaking from Woking in Surrey, she added: "At the moment there is no other process or procedure ongoing to tell us how the bombing was carried out, why it was done, the motivation for it and who ordered it."
Martin Cadman, 84, of Burnham Market, Norfolk, who lost his son Bill, 32, said: "It's been nearly 21 years since the event and where are we? Nowhere."
Meanwhile, Daniel Kawczynski, the Tory chairman of the all-party parliamentary group on Libya, said Megrahi should not be freed until the killer of Pc Yvonne Fletcher Â who was shot dead outside the Libyan embassy 25 years ago - was handed over.
The Lockerbie memorial
Abdelbaset Ali al-Megrahi was revealed today to have requested his second appeal to be dropped. Al-Megrahi was convicted of planting a bomb on Pan Am Flight 103, which detonated as the aircraft flew over Lockerbie, Scotland in 1988 killing all 259 people on board and eleven more on the ground.

Megrahi had been offered transfer to his home of Libya if he dropped the appeal, but had previously stated he would not return unless his conviction was overturned. He has terminal prostate cancer and it is thought that he is due to be released shortly on compassionate grounds, having served eight years of his minimum term of 27 years in Greenock prison.

Megrahi lost his first appeal in 2002, but a review found that there was reason to believe there may have been a miscarriage of justice. The decision to drop his second appeal comes as a surprise, with some victims' relatives expecting the proceedings to continue even after his death. A court hearing to consider the request is due on Tuesday.

Opinions about the conviction are split, with the families of most British victims considering the man to be innocent while those in the United States are satisfied with his guilt. Briton Martin Cadman, whose son Bill was among the dead, told ''The Daily Telegraph'' "It's been nearly 21 years since the event and where are we? Nowhere." Last year, on the twentieth anniversary of the disaster the head of the US group Victims of Pan Am Flight 103, Kara Weipz, told ''Wikinews'' that "There is no difference between the truth as we see it and the official version of events. The facts are the facts, Mr. Megrahi is guilty."