The abductions on Thursday in the Sinai Peninsula prompted angry police to shut down border crossings with Gaza and Israel, piling the pressure on Mursi to secure the hostages' release.
"There are no negotiations with criminals and the awe of the state will be preserved," Mursi was cited as saying by the official MENA news agency.
A video posted on YouTube today appeared to show the seven hostages, blindfolded and with their hands on their heads, identifying themselves.
He mentioned by name a Bedouin militant sentenced to death after a 2011 attack on a police station in the north Sinai town of El-Arish.
"We hope that you, president, quickly release the political activists from Sinai as soon as possible because we can no longer stand the torture," said one hostage.
The video was posted online by an anonymous account.
One of the hostages in the footage bears a strong resemblance to a hostage in a picture held by relatives who joined the police protest at the Rafah border crossing.
Attacks on police and soldiers in the sparsely populated peninsula have surged since an uprising that overthrow president Hosni Mubarak in 2011, as have cross-border attacks on Israel.
The north of the peninsula is underdeveloped and has become a haven for Islamist militants, unlike the south which is dotted with beach resorts.
In August 2012, suspected Islamist gunmen killed 16 Egyptian soldiers near Israel's border and commandeered an armoured vehicle into Israel, where they were stopped by a helicopter strike.
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