(AhlulBayt News Agency) - Mr Mubarak also dismissed his government and said a new cabinet would be announced on Saturday.
It was his first statement since the protests - in which at least 26 have died with hundreds injured - began.
Tens of thousands took part in protests in Cairo, Suez, Alexandria and other cities.
Protesters set fire to the headquarters of the governing NDP party and besieged state TV and the foreign ministry.
At least 13 people were killed in Suez on Friday, while in Cairo, five people died, according to medical sources.
That brings the death toll to at least 26 since the protests began on Tuesday.
"I have asked the government to present its resignation today," Mr Mubarak said, adding that he would appoint a new government on Saturday.
He also said he understood the protesters' grievances but that a thin line divided liberty from chaos and he would not allow Egypt to be destabilised.
One protester told the Associated Press news agency: "We want Mubarak to go and instead he is digging in further."
In a televised address shortly after Mr Mubarak spoke, US President Barack Obama said he had spoken at length with the Egyptian president and urged him to turn "a moment of volatility" into "a moment of promise".
The BBC's Jon Leyne in Cairo says there had clearly been a lot of discussion behind the scenes before Mr Mubarak spoke to the country.
But his comments will probably just provoke further unrest, says our correspondent - the people on the streets will be both infuriated by his accusations that they are seeking to destabilise the country and inspired that, having wrung some concessions from him, they could yet manage to oust him.
After Mr Mubarak spoke, a sustained volley was heard from central Cairo, which our correspondent said could have been either tear gas or live fire.
The Reuters news agency later quoted witnesses as saying more than 20 military vehicles rolled in to central Tahrir Square shortly after midnight, scattering protesters into the sidestreets.
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