Opponents of Mohamed Mursi, the first elected head of state in Egypt's history, staged more street protests in Cairo, while his supporters in the Muslim Brotherhood held funerals for six of the movement's members killed in fighting around the presidential palace earlier in the week.
Mr Mursi had offered few concessions in a speech late on Thursday, but refused to retract a decree in which he assumed sweeping powers or cancel a referendum next week on a constitution drafted by an Islamist- dominated assembly.
Instead, he called for a dialogue at his office to chart a way forward for Egypt after the referendum, an idea that liberal, leftist and other opposition leaders rebuffed. They have demanded that Mr Mursi rescind the decree in which he temporarily shielded his decisions from judicial review.
A leader of the main opposition coalition said it would not join Mr Mursi's dialogue: "The National Salvation Front is not taking part in the dialogue," said Ahmed Said, a leader of the coalition, who also heads the liberal Free Egyptians Party.
The Front's co-ordinator, Mohamed ElBaradei, a Nobel peace laureate, urged "national forces" to shun what he called an offer based on "arm-twisting and imposition of a fait accompli".
Mr Mursi's decree giving himself extra powers sparked the worst political crisis since he took office in June and set off renewed unrest that is dimming Egypt's hopes of stability and economic recovery after nearly two years of turmoil following the overthrow of Hosni Mubarak.