Last week, Swansea East Labour Party held a vote in which members voted unanimously in favour of a resolution confirming support for Mr Corbyn.
Pontypridd MP Owen Smith branded himself as the "radical and credible" alternative to Jeremy Corbyn as he launched his bid for the Labour leadership.
He continued: "One instant policy difference is that I have come up and said there should be a second referendum to give people a chance to test that".
Corbyn said he was "delighted" at the result of the secret vote, which backed him by 18 votes to 14, and emerged triumphant from the talks to cheers from supporters.
A statement released by the constituency party says: "This constituency expresses deep concern over the apparent disregard shown by some members of the Parliamentary Labour Party (PLP), towards democracy in our party".
He said the leadership campaign, in which he would fight "on all the things that matter", from inequality and poverty to improving opportunities, would strengthen Labor.
Smith will need the support of 51 MPs or MEPs to be eligible to stand in the contest.
Mr Corbyn has dismissed the claims, insisting he is "going nowhere" and intends to honour the mandate given to him by 59.5% of party members in last September's leadership election.
Late Tuesday, Corbyn won a first victory over his critics after the party's executive committee ruled he would automatically be included on the leadership ballot.
Meanwhile, Eagle had announced her candidature on Monday and has since had her constituency office in Merseyside region of England vandalised.
The union also overwhelmingly voiced support for Mr Corbyn as he faces attempts to try to oust him.
A Jewish Labour donor and former party candidate is preparing later today to legally challenge the party's decision to automatically place Jeremy Corbyn on the leadership ballot.
And only 23% think the current leader has what it takes to be prime minister but just 21% think Ms Eagle would make a good PM.
"The Parliamentary Labour Party needs to get behind the leader.
None has been undertaken by any Labour Party member as far as I am aware, and I deplore such behaviour, and would never condone that, but having said that, it is not right to conflate abuse and intimidation with refusing to call MPs to account".
He added that the present situation where most MPs had lost confidence in their leader "forces people like me to put our hats in the ring to say I can heal the party".
"I welcome the contest ahead", she said on Twitter.
Dame Margaret accused "people around Jeremy" of indulging in "a politics of intolerance, bullying and intimidation", including anti-Semitic emails she had received and reported to the police.