The veteran socialist has clung on even though three-quarters of the party's lawmakers say they have lost confidence in his leadership and Welsh MP Owen Smith has launched a challenge to oust him.
"It's the job, it's the duty, it's the responsibility of every Labour MP to get behind the party," Corbyn said at the start of his leadership campaign.
"I hold out the hand of friendship," he said, addressing "those who may not agree with me politically, may not even like me personally".
Some commentators have warned the outcome could split the historic Labour Party, which grew out of Britain's trade union movement in the 19th century.
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The 67-year-old Corbyn is automatically on the ballot as the current leader and he has strong grassroots support but Smith is backed by 162 of the party's 230 MPs including former leader Ed Miliband.
Some 180,000 Labour supporters have signed up to vote in recent days.
"We, the Labour party, will be stronger, even stronger, and hopefully even bigger at the end of this campaign," Corbyn said.
Corbyn critics -- many of them supporters of the more centrist policies of former leader Tony Blair -- accuse Corbyn of pursuing a leftist agenda that could keep the party out of power for years.
The 46-year-old Smith, who previous worked as a BBC producer and a lobbyist for pharmaceuticals giant Pfizer, has paid tribute to Corbyn for "helping Labour discover its radical roots" but said the party needed a new generation to take it forward.