The students, arrested in August and charged with conspiring to kill Israeli soldiers, were handed over on Sunday at the Egyptian consulate in Tel Aviv and arrived in Cairo later, Egyptian security sources said.
Azzam is an Israeli Druze businessman who was found guilty of spying in 1997 and was imprisoned for 15 years.
Azzam crossed into Israel from Egypt's Sinai peninsula at the Taba checkpoint near the Israeli Red Sea resort of Eilat. "He was very excited and screamed 'I am born again'," a witness said.
Israel agreed to release the students when Egyptian Foreign Minister Ahmad Abu al-Ghait and intelligence chief Umar Sulayman visited Israel last week, the official Egyptian al-Ahram daily said.
Another state newspaper, al-Jumhuria, said on Saturday Egyptian President Husni Mubarak had intervened with Israel on behalf of the students.
Egyptian FM Abu al-Ghait
discussed the matter with Israel
Mubarak told reporters on Saturday that Foreign Minister Ahmad Abu al-Ghait and intelligence chief Umar Sulayman raised the issue with Israeli officials during a visit on Wednesday.
Israeli authorities say they arrested the students near the desert border between the two countries for allegedly crossing into Israel illegally, armed with an airgun and knives.
State prosecutors told a magistrate's court in the Israeli town of Beersheba the students had plotted to kidnap Israeli soldiers to use as bargaining chips for Palestinian prisoners.
The students are also accused of having intended to hijack an army tank, kill its crew and then rob a bank to get money that could be used to fund further attacks on Israel, according to Israeli state prosecutors.
Parents of the students have insisted their sons had no political views or affiliations and had gone to Israel merely to find work, as many young Egyptians have done in the past.
Al-Ahram named the students as Imad Sayyid, Muhammad Yusri, Mustafa Mahmud Yusuf, Mustafa Abu Daif, Mahmud Jamal Izzat and Muhammad Mahir.
The students are aged between 21 and 25, the paper said.
Relations between Cairo and Tel Aviv were strained by the arrests and the killing on 18 November of three Egyptian border policemen by Israeli tank fire on the Egyptian side of the border.
The killing of the border policemen
strained Egypt-Israeli relations
Cairo lodged a formal complaint after Israel said the army killed the policemen by "mistake", threatening to plunge their delicate diplomatic ties into crisis mode.
Israeli Prime Minister Ariel Sharon is said to have defused the row by calling President Husni Mubarak to express "deep sorrow" over the incident on the Gaza border and promised to keep him abreast of an official investigation.
There has been no report yet on the findings of the inquiry. The families of the victims are reported to suing both Egypt and Israel for the killings.
According to Haaretz newspaper the Egyptian government is said to have paid each of the families 70,000 Egyptian pounds ($11,300) in compensation.