A military official speaking on condition of anonymity confirmed the invitation from Sissi, who is also defence minister, to AFP.
He said the appeal was made to leaders of opposition coalition the National Salvation Front and other dissident figures, as well as to Islamist groups that back President Mohamed Morsi, notably the Muslim Brotherhood's Freedom and Justice Party and hardline Salafist groups.
The proposed venue for the talks is a military sports complex in northeast Cairo, he said.
Sissi made his call as rival protesters gathered in Cairo for mass rallies for and against a constitutional referendum called by Morsi for Saturday.
The army has orders from Morsi to use temporary police powers to protect state institutions, following clashes in the capital last week that left seven people dead and hundreds hurt.
The army, which ruled Egypt between the February 2011 ouster of autocratic leader Hosni Mubarak and the June 2012 election of Morsi, has tried to remain neutral in the crisis but has warned it will not allow the situation to worsen.
At the weekend, the military demanded talks between the two sides, but the opposition said no dialogue was possible unless Saturday's vote is called off.