Mr Smith said the leadership contender with the most support among Labour MPs should become the unity candidate to take on Mr Corbyn.
But Ms Eagle refused to give the same assurance, instead insisting that she is in the best position to beat Mr Corbyn to become leader in a national ballot of members.
The comments come ahead of a PLP hustings on Monday night where the leadership contenders will set out their stall to fellow MPs.
Asked about the leadership race on the BBC’s Andrew Marr Show, Mr Smith said: “I think one of us standing would be better, is the honest answer, but I think the PLP has got to be a grown-up organisation and try to come to a decision in the next couple of days over who it is.”
Asked how the pair would decide who commands the most support and take on Mr Corbyn, he suggested the candidate with the largest number of nominations could be the challenger.
He said: “That (nominations) is one way in which we can do it, or we can make an agreement between ourselves.
“But my view is whoever is the person who commands the largest degree of support in the PLP is the unity candidate, and that’s the person who should go forward and take Jeremy on.”
But Ms Eagle, who launched her leadership bid last week, refused to sign up to the idea that the contender with the fewest nominations should step aside.
She said it is “too early” to talk about the issue and told the show’s host “we are not going to do a deal here on your sofa”.
She later added: “I think we have to have the person who is most likely to beat Jeremy Corbyn, and I think that is me.”
It comes as senior Labour MPs called for a single unity candidate, warning that a protracted and divisive leadership battle would detract from challenging the new Prime Minister Theresa May.
Former home secretary Yvette Cooper told The Guardian: “Angela and Owen are both really talented and could do the job. But I don’t think it would help to have an artificial battle between the two of them when they agree about so much.”
There is mounting speculation that Labour could split if Mr Corbyn clings on to power despite having been abandoned by most of his MPs.
But Mr Smith, the former work and pensions secretary, ruled out leaving the party.
He said: “Look, I’m not leaving the Labour Party, not for Jeremy Corbyn or for anybody else. I’ve always felt it is the Labour Party or nothing for me.
“But I will serve and work with Jeremy if he wins.”
However Ms Eagle slammed the current leadership and did not give any indication she would return to Labour’s frontbench if Mr Corbyn survives.
She said: “Jeremy’s lost the confidence of the Parliamentary Party and we have seen from some of the polls that he is not connecting either with Labour voters, and that is the argument we have to have in the country.
“And we may have an early general election so let’s do the leadership election and let’s see what happens.”