The former shadow cabinet member, who stepped down amid a slew of resignations last month, is a second challenger, raising speculation that his entry could benefit Corbyn.
Smith told BBC Radio 4 this morning that Corbyn is a "good man with great Labour values", but is "not a leader", adding that he can heal the party.
"I wanted this to be resolved without a damaging, divisive leadership contest. I wanted Corbyn to find a way to heal the Labour party," Smith said, adding he was not willing to let the party split.
"I would be a radical and credible leader of the Labour party and I can heal this party."
Yesterday Labour's National Executive Committee (NEC) ruled that Corbyn would be able to stand without having to garner 20 per cent of support from the Parliamentary Labour Party (PLP) of European Parliamentary Labour Party, equivalent to 40 MPs.
The NEC voted voted 18-14 in favour of letting him stand.
The decision yesterday followed a bid by former business secretary Angela Eagle in the wake of a no confidence vote by the PLP in Corbyn.
A timetable for the contest will be announced tomorrow.
While yesterday was a big win for Corbyn and the left of the party, the rebels in the party have a silver lining in that the rules for electing the leader have changed.
Last year Corbyn was voted in by a mass of grassroots voters, many of whom had registered to vote for just £3.
Now, that price has increased to £25 and must be paid during a two-day window next week, while full-time members must have been affiliated with the party for at least six months to vote.
Momentum, which was set up following Corbyn's Labour leadership election last year to garner enthusiasm in the grass roots for the left-wing of the party, said the decision to change rules by which people can vote is "not fair".