War veterans until recently loyal to the 93-year-old president and liberal groups have also urged him to quit.
Earlier Mr Mugabe made his first public appearance since the takeover.
Mr Mugabe had been under house arrest for days, but attended a graduation ceremony on Friday, handing out degrees.
The army made its move after a power struggle over his successor. Mr Mugabe sacked Vice-President Emmerson Mnangagwa last week, apparently to pave the way for his wife Grace Mugabe – who is four decades younger than him – to take over the presidency instead.
The military said it was “engaging” with Mr Mugabe and would advise the public on the outcome of talks “as soon as possible”.
At least eight out of 10 regional party branches voted on Friday evening for Mr Mugabe to resign as president and party secretary.
In an unprecedented broadcast, several regional leaders appeared on state TV saying that he should quit.
They also called for Grace Mugabe to resign from the party, and for Mr Mnangagwa to be reinstated in the party’s central committee.