When contacted for comment yesterday, Mr O'Dowd said the National Party "meets each and every Monday in the party room during sitting weeks".
Also, the ABC reported today that Joyce faces fresh allegations of sexual harassment.
Before Mr Joyce's announcement, Dr Gillespie said he was a safe pair of hands with ministerial experience and a record of delivering in his electorate.
"I have asked that that be referred to the police", he said.
As pressure on his position grew, he did press interviews in which he blamed public life for the breakdown of his marriage.
Ministers' relatives and partners are not allowed to be appointed to executive government offices without approval from the prime minister, but Joyce battled the calls for his resignation by claiming Campion was not technically his partner at the time she was appointed to the various roles. Australia's High Court deemed Joyce was a New Zealand citizen in October, forcing Turnbull to rule in minority until Joyce re-contested and won his seat again.
During the election campaign, despite eventually enjoying a positive swing, he faced a number of security alerts and media speculation about his personal life.
Nationals President Larry Anthony, the party's senior bureaucrat, said in a statement: "The party will greet this news with a heavy heart but we understand and respect his decision to stand down as leader". "Over the last half a month, a litany of allegations and might I say right here ... the leaking background will destroy my government - it will destroy any government..."
It was 10pm in Washington DC when Mr Joyce announced his resignation from the front bench and the party leadership, and he still had not directly spoken to the Prime Minister on his decision.
WA leader Mia Davies said Joyce's leadership was no longer tenable.
"It's got to stop".
Mr Turnbull also announced a ban on sex between ministers and their staff and launched an extraordinary attack on Mr Joyce's relationship - calling it "a shocking error" that unleashed "a world of woe".
The Coalition between the Liberals and the Nationals is Australia's most successful political partnership, having endured for more than 95 years.
Veterans Affairs Minister Michael McCormack and David Gillespie, the assistant minister for children and families, both said they would contest the leadership ballot on Monday.
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