The coalition "calls on all revolutionary forces and political parties and patriotic figures to enter a deep dialogue on exiting the current crisis," it said in a statement.
The coalition, which has organised weekly protests despite a harsh police crackdown, insisted in its statement on keeping up "peaceful opposition" but said it wanted a "consensus for the public good of the country."
The proposal is the Islamists' most flexible yet made in public, and comes "with no conditions," a coalition official said.
Much of the Brotherhood's leadership has been put on trial, including Morsi himself.
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"We have no conditions, and neither should they," Imam Youssef, a leader of the Asala party, which is part of the Islamist coalition, told AFP.
But he added the talks must lead to a "democratic" solution, and the coalition wanted them to start within two weeks.
The coalition was prepared to discuss "all solutions that lead to stability."
The Islamists were prepared to respect the demands of the millions of protesters who took to the streets calling for Morsi's ouster, Youssef said.
Asked if the coalition would insist on Morsi's return to office, he said: "We don't want to get ahead of ourselves."
The Islamists proposed conditions to enter the dialogue, including the release of Islamist prisoners and the reopening of Islamist broadcasters shut down after Morsi's July 3 overthrow by the military.