Shadow chancellor John McDonnell expressed concern at reports in the Daily Telegraph that some MPs opposed to Mr Corbyn are preparing to elect their own leader and launch a legal battle for the party’s name and assets if Mr Smith loses the membership ballot election.
“The democratic process is fundamental to Labour Party values and all candidates must commit to respecting the outcome of this election.
“We call on Owen Smith to condemn the minority of MPs supporting his campaign who are threatening to subvert the outcome of this election and cause enormous damage to the Labour Party,” Mr McDonnell said.
Responding to the intervention, Mr Smith said he would “not indulge in gossip”.
“This leadership process is now entirely about Labour members and supporters. We have a one member, one vote system, so MPs will have the same amount of say as all Labour members - exactly as it should be,” he said.
The spat came as Mr Smith takes his battle to topple Mr Corbyn into “enemy territory” with a campaign push in Liverpool.
With the Smith camp seeing the city as a Corbyn stronghold, they want the move to underline the would-be leader’s aim to try to reunite the party around him.
The bid comes as Mr Corbyn holds rallies in Hull and Leeds as the two men slug it out for the Labour crown through the summer before the winner is declared at the end of September.
The pitches came as former shadow cabinet member Angela Eagle warned Labour MPs to moderate their language after Mr Smith was forced to apologise for controversial comments aimed at Prime Minister Theresa May.
Mr Smith, who toppled Ms Eagle as the “unity” candidate tasked with ousting Mr Corbyn as Labour leader by dissenting MPs, drew widespread criticism for saying he wanted to “smash” Mrs May “back on her heels”.
Ms Eagle told The Guardian: “He should have, and has, apologised. As someone that has heard cruel words spoken about me, I know that language matters. And we’ve all got a responsibility to be sensitive with our use of language.
“Owen has shown a capacity to recognise, and apologise for, insensitivity, and that’s important.”
Ms Eagle said it was “extraordinarily disingenuous” for Mr Corbyn’s team to “cloak Jeremy in the mantle of feminism by saying that we have to be sensitive about our use of language, especially around women MPs”.
She insisted she was not taking it personally that more anti-Corbyn MPs had backed Mr Smith over her as the best placed candidate to dislodge the leader.
The former shadow business secretary said she was trying not to dwell on threats received since she became the first to challenge for the leadership.
“I don’t want to dwell on it, partially because if I dwell on it too much, or I talk about it too much, I just get a load more,” she said.
Mr Smith initially refused to apologise for his comments regarding Mrs May, but was forced to do so after the controversy mounted.
The visit to Liverpool by Mr Smith sees him attending the city’s gay pride celebrations, and meeting party members at a special event.