Eagle, who had resigned as shadow business secretary last week, said Corbyn had failed "to lead an organised and effective" party.
Confirming her leadership bid today, Eagle said she would "explain my vision for the country and the difference a strong Labour Party can make" on Monday.
Corbyn's spokesperson said he would remain as leader of the party and would fight any leadership challenge.
"[He] is ready to talk with as many people as necessary to assist that process, discussing policy initiatives and listening to ideas," he said.
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"I urge all my colleagues to listen very carefully to them and indeed come together to oppose what this government is doing to the most vulnerable within our society," Corbyn said at a miners' event earlier this week.
Watson said Corbyn's declaration to continue "come what may" meant there was "no realistic prospect of reaching a compromise" over the leadership.
More than 20 members of Corbyn's shadow cabinet and a similar number of junior ministers walked out last month, after questioning his performance during the EU referendum and his ability to lead the party.