Opposition leaders earlier rejected Mr Morsi's call for talks, putting their weight instead behind rallies which converged on the palace from all across Cairo attacking his plans for a constitutional referendum next weekend.
The protesters were held up by a line of the Republican Guard, but as numbers grew it gave way, with swarms of demonstrators running in front of the gates and climbing on top of armoured personnel carriers and tanks.
In a long-awaited televised speech to the nation, Mr Morsi refused to lift the declaration under which he put his powers beyond the scrutiny of judges and insisted the referendum on a new, Islamist-tinged constitution would not be postponed.
He called for a meeting with the opposition on Saturday, but his failure to offer compromises in advance, and the increasingly militant tone of Brotherhood statements, infuriated the mainly liberal and secular opposition.
"We are against dialogue based on a policy of arm-twisting and imposing a fait accompli," said Mohamed El Baradei, the former United Nations atomic energy chief who is now the opposition's figurehead.
The Brotherhood held a funeral at Al-Azhar mosque for two of those killed in clashes outside the presidential palace on Wednesday evening, when thousands of Brotherhood and opposition supporters fought.
Speakers accused the opposition of being murderers, traitors and "hash-smokers", a reference to drugs and alcohol they alleged the opposition were using. Mohammed El Beltagi, once seen as a moderate Brotherhood leader, said: "We will keep going even if we all become martyrs. We will avenge them or die like them."
The crowd chanted: " Egypt is Islamic, it will not be secular, it will not be liberal."
The marches on the palace attracted supporters who had not previously joined in demonstrations, including many of Cairo's wealthy, secular elite who fear a threat to their traditionally secular lifestyle.
There were more unveiled women in the crowds outside the palace than at any recent gathering in the capital. (© Daily Telegraph London)