These things, according to Labor MP Deb O'Neill, must be cleared by Mr Abbott himself, adding the Liberal chief needs to prove without doubt that "he's good enough to be prime minister of this country."
"I call on the leader of the opposition to make a statement to the parliament in relation to (these) matters," Ms O'Neill told reporters in Canberra on Thursday.
She joined earlier declarations by Leader of the House Anthony Albanese, who on Monday urged Mr Abbott to clarify the accounts detailed by journalist David Marr in The Quarterly Essay.
Mr Marr wrote in his essay that Mr Abbott had intimidated a fellow student leader during his Sydney University days, by punching the wall where Barbara Ramjan was leaning following his defeat during a 1977 university election.
There has to be a full explanation on the part of the Coalition leader, Mr Albanese said, adding "it is Tony Abbott himself who says that it's legitimate to go back into people's past decades."
The Labor MP was referring to earlier calls from the opposition that Prime Minister Julia Gillard must explain her deeds while acting as lawyer for a labour leader suspected of corruption more than a decade before becoming PM.
Ms Gillard did and it's now the turn of Mr Abbott, Labor MPs said, in light of new stories published by Fairfax Media on Thursday that largely supported what Mr Marr had written and Ms Ramjan had shared.
In his defence, Mr Abbott told Mr Marr that for the incident to be true "it would be profoundly out of character," insisting that he could not recall the intimidation episode as described in the essay.
He later said this week that the incident did not happen at all.
But Finance Minister Penny Wong said today that in order to clear the air, Mr Abbott needs to do what he demanded of others, especially his opponents.
"Tony Abbott is the person who has been telling Australians and journalists for months that leadership is also about character, and that issues about your past are relevant to character," Ms Wong told Sky News on Thursday.
Defending the opposition leader, Deputy Liberal Leader Julie Bishop told The Australian that what Labor MPs have been saying lately were but expected and warned: "We will see an unrelenting campaign by the Labor Party and their sympathisers on the Left against Tony Abbott."
Also, National Leader Barnaby Joyce is under the impression that Mr Abbott hardly needs to say anything to dispel the controversies created by Mr Marr's reporting, which Senator Joyce described as biased.
"You are only going to get one point of view from David Marr . . . he absolutely detests anyone on the coalition," the National senator was quoted by the Australian Associated Press (AAP) as saying.
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