The presiding judge in the trial asked the mufti -- the country's official interpreter of Islamic law -- to consider death sentences for the six codefendants, saying the court would convene again on June 18 after the mufti's response.
It will then pronounce its verdict and sentence for the remaining five defendants, including Morsi, on charges of having supplied Qatar with classified documents.
Egyptian law requires the mufti to sign off on death sentences. His opinion is not binding but is usually respected by courts.
Three of the defendants sentenced to death were tried in absentia and identified by the prosecution as journalists who helped relay the documents to Qatar.
Qatar was a main backer of Morsi and his Muslim Brotherhood movement while he was in power between 2012 and July 2013, when the military overthrew and detained him.
He has already received sentences in three separate trials.
He has been sentenced to death for his alleged role in prison breaks and attacks on police stations during the 2011 uprising that overthrew veteran strongman Hosni Mubarak.
Morsi was one of the leaders of the Brotherhood who had been jailed during the 18-day uprising, escaping with thousands of inmates who broke out of prison.
He was also sentenced to life in prison for espionage on behalf of Iran and militant groups Hezbollah and Hamas, and received a 20 year term for deadly clashes outside the presidential palace in 2012 between his supporters and opponents.
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The Qatar espionage case stems from allegations that aides passed on classified state secrets to Doha using intermediaries.
The six facing a death sentence include Ibrahim Mohamed Hilal, whom the prosecution had identified as a senior editor with the Qatar-based channel Al-Jazeera.
He is not in Egypt and was tried in absentia.
An Al-Jazeera official said he is "an adviser" to the channel's chairman.
Another, also tried in absentia, is Jordanian citizen Alaa Omar Mohamed Sablan, identified by the prosecution as an Al-Jazeera producer.
Asmaa Mohamed al-Khatib, identified as a reporter with the pro-Brotherhood Rassd news outlet, was also sentenced to death in absentia.
They were alleged to have assisted in relaying the classified material to Doha.
Egypt had cracked down on Al-Jazeera and imprisoned three of its journalists in Cairo, including Australian Peter Greste and Canadian Mohamed Fahmy, before releasing them by presidential decree.
The defendants will be able to appeal the rulings before the court of cassation, and those tried in absentia can win a retrial if they hand themselves in.
Egyptian courts have sentenced hundreds of Morsi supporters to death since his overthrow, but many have appealed and won new trials.
A police crackdown on his supporters has killed hundreds of protesters, while a jihadist insurgency has killed hundreds of policemen and soldiers.