Ms Eagle, who resigned as shadow business secretary last week, said leader Jeremy Corbyn had failed "to lead an organised and effective" party. Mr Corbyn's spokesman insisted he would remain as Labour leader and would fight any leadership challenge.
It comes after Labour's deputy leader Tom Watson called off talks with unions aimed at resolving the stand-off.
Unite's general secretary Len McCluskey, who had been leading union efforts, said Mr Watson's actions were an "act of sabotage", and hinted at a possible split in the party.
Mr Corbyn has refused to step down despite a vote of no confidence passed by the party's MPs last month - and many resignations from his shadow cabinet - in the wake of the EU referendum.