The petition was started a week ago by Brisbane high school student Corbin Duncan, collecting 500 signatures in the first 30 minutes.
It calls on Prime Minister Turnbull to take a $1 a year salary or donate his $500,000-plus salary to charity.
He said he was inspired by the example set by others in taking on a $1 a year salary, with adoptees of the practice including businessmen Elon Musk of Tesla, Mark Zuckerberg of Facebook; and former US president John F Kennedy.
Last week, US presidential candidate Donald Trump said he would refuse the $AU570,000 annual salary that comes with the White House job, if he is elected.
"Make no mistake, you deserve and are entitled to the Prime Ministerial compensation package, however I am sure you have not taken on the role of PM for the associated salary," the petition reads.
"As a 15-year-old Australian, with a great interest in politics and current affairs, I implore you to consider my request. By taking a $1 salary, you will show your fellow Australians, you are truly a servant of our great country and its citizens."
The proposal received enthusiastic support from those who signed the petition.
"Not that I think you don't deserve to be paid for the top job, or anything, I think this kid has a good point," wrote Clint Mackay in Brisbane.
"You'll still be able to pay your bills and enjoy the same lifestyle without the salary so why not put that money where the need is greater? Make us proud Malcolm!!!"
Jason Cotton said he supported the idea because it would set a strong example.
"This would be a sign of genuine dedication to the people to represent without fear or favour from a person who has the ability to support themselves financially. A nation of lifters and not leaners with an example from the highest level. A charity would benefit immeasurably from your gift."
Samuel Yeo said the gesture would change people's perception of political leaders.
"Given Turnbull's personal wealth and his forward-thinking mindset, Corbin's suggestion is a really impressive way for our country to rejig its view of politics into a duty, a service, a position of leadership."
Earlier this week, Australia’s 29th prime minister said he will reside in his own waterfront property at Point Piper over the official prime ministerial residence of Kirribilli House.
"It's very valuable, it's a great location to use for charities and for opportunities to support good causes. So we'll certainly be doing that. But we'll be sleeping at home," he told ABC television.
The former merchant banker and lawyer built his wealth when he parlayed $500,000 investment in internet provider OzEmail into $60 million sale more than a decade ago.
Considered the second-richest parliamentarian after Clive Palmer, the BRW's Rich List in 2010 estimated Mr Turnbull's wealth to be $186 million.
More than 2000 people have signed the petition.
