PARIS (AP) _ The East German government made a formal request to Syria for the extradition of Nazi war criminal Alois Brunner, Adolf Eichmann's deputy during World War II, a French Nazi hunter said today.

Syria has consistently denied Brunner's presence on its soil, although he has been photographed and interviewed at his home in Damascus.

The extradition request was presented in Damascus today by East German Ambassador Karl-Heinz Lugenheim, Nazi hunter Serge Klarsfeld said.

An East German Foreign Ministry spokesman, who spoke on condition of anonymity, confirmed in East Berlin that Lugenheim had been instructed to request the extradition. The spokesman gave no further details.

Brunner, 78, who has been living in Dasmascus since 1955, is considered the most important Nazi criminal still at large.

According to the World Jewish Congress, the Austrian-born Brunner is responsible for the deaths of more than 125,000 Austrian, French, Greek and Czechoslovakian Jews. Hitler assigned Eichmann to prepare ''The Final Solution,'' the extermination of European Jewry.

After the war Brunner was detained by American and British officials but kept his identity secret.

He was tracked down in Syria under the name Georg Fischer in the late 1950s by Austrian Nazi hunter Simon Wiesenthal and officially identified by Serge Klarsfeld and his wife, Beate, in 1982.

The Klarsfelds have lobbied actively for Brunner's extradition. In January, Serge, a Paris lawyer, was expelled from Syria after planning to hold a public debate on Brunner.

Beate has devoted her life to bringing war criminals to justice. In February, she met with East German prosecutor Horst Busse to discuss the Brunner affair.

After 40 years of denial, East Germany recently admitted its responsibility in Nazi war crimes, formally apologizing to the world Jewish community and announcing it would pay reparations. East Germany joins France, Austria and West Germany in requesting Brunner's extradition.

Brunner has been photograhed and even interviewed at his heavily guarded home in Damascus. In 1985, Brunner was photographed by Bunte, a West German news magazine, and showed no remorse for his crimes.

''I'm not sorry I killed those vermin,'' he was quoted as telling interviewers.

Two years later, he told an American newspaper in a telephone interview that the Jews deserved to be killed because they were agents of the devil. He said if he had to do it over again, he would.

Brunner was sentenced to death in absentia twice in 1954, in Paris and Marseille, for taking part in the deportation of more than 120,000 Jews from France to German death camps.

Mrs. Klarsfeld has said the Israeli secret services also found Brunner, and were probably behind the letter bombs which cost him an eye and several fingers.