Opinion polls show most Australians support gay marriage.
"I'm a strong leader", the prime minister told reporters in Canberra, sticking by the plebiscite policy he took to the 2016 election.
The government blocked a national vote last October and said it would not introduce a Rebel MP's bill on same-sex marriage.
ABC News 24 reported response for postal votes was at about 13-15% and susceptible to voter fraud.
Specifically, he suggested the government could compromise by ditching the $15m of public funds for each of the yes and no case in the plebiscite "because there's been so much debate it may not be needed", and even consider "what the bill would look like". This is expected to be rejected by the Senate.
Mr Entsch told The Courier-Mail last night he was willing to let the plebiscite process play out, if it was given a definitive time frame, but he "reserved his rights" with regards to a free vote.
Mr Turnbull is a personal supporter of same-sex marriage, but during his own leadership bid gave private assurances to his party's anti-LGBT lobby that he would stick by the plebiscite plan.
He blasted both sides of politics, saying Opposition Leader Bill Shorten was "determined to wedge (Malcolm) Turnbull to ensure marriage reform is delayed until he himself is the one who gets to sign off on it".
"There are arguments against plebiscites, I understand that".
The Coalition plans to get the original plebiscite before the Senate again before the week is out.
Federal cabinet is meeting Monday to rubber stamp a proposal by Senator Cormann to put to the Liberal Party meeting later in the day in which the party will adopt as policy a postal ballot in the likelihood the Senate again rejects legislation to establish a plebiscite. "The only reason it has not been held is because of (Labor leader) Bill Shorten's opposition".
The marriage equality movement is opposed to the postal vote solution, and has warned it may test it in court.
"We should just get on and have a vote on marriage equality straight away in the Parliament", he added.
"As I've said before in, I will be voting yes and I'd encourage others to do so".
Sheldon handed the petition to Sen.
But Anne Twomey from the Sydney University Law School believes this could be done legally by a regulation under the Financial Framework (Supplementary Powers) Act - although the regulation could be disallowed by the Senate. But most of that support is soft, and gay-rights advocates are concerned that referendums rarely change the status quo in Australia.
The urgency and timing of the meeting reflect the fact that the divisions over marriage equality are bound up with deeper and broader rifts wracking the fragile government and the entire parliamentary establishment.
Vicki Moody-Basedow designed the couple's white costumes, embroidered with "righteousness", "holiness" and "Jesus".
She noted that some MPs have said they would ignore the result.
Most Liberal MPs have called for the issue to be resolved swiftly to try to end a debate that has exposed the party's bitter internal divisions.