A few politicians have publicly condemned the practice, but leaders in the Muslim communities, trying to soothe the debate, say not all Muslim males avoid such handshakes.
The debate ignited after revelations three Australian universities advise students to respect that some Muslim men do not shake women's hands.
It was also revealed at least two public schools in Sydney's west allow students to refuse.
Coalition MP George Christensen called the practice "unAustralian," and former House of Representatives Speaker Bronwyn Bishop claimed Muslim men deem women "unclean."
The vice-president of the Islamic Council of Victoria, Adel Salman, says that simply is not true.
"That's absolute nonsense, and it's actually appalling -- absolutely appalling -- that someone of that standing, a former senior minister, would actually make that statement. That is absolutely appalling. It's got nothing to do whatsoever with the state of hygiene of the male or the female."
Some Islamic scholars say there should be no unnecessary physical contact between men and women who are not married or related by blood.
Ahmadiyya Muslim Community imam Inam al-Haq Kauser has told SBS not all Muslims adhere to that belief.
"There's nothing in the Holy Koran mentioning about not to shake hands. It is actually in the borderline area. To me, if any lady extends her hand, we must shake hands with her, because, otherwise, it will be very insulting and it will be very offensive. I think the shaking hands is no problem."
Adel Salman says the practice comes from interpretations of Islamic scripture known as the Hadith. (hah-DEET)
"I don't think you can pinpoint one particular Hadith and say, 'That's it. That's the one that says there shall be no handshakes between them.' That's not the way to do it. The reason for that is maintaining that sense of modesty between the sexes. Look, I think culture is definitely part of it. And I should be very clear that the majority of scholars would say that there should be no physical contact between men and women. Having said that, there are some scholars who say it's fine as long as it's done respectfully, if it's done with modesty, it's not done with any other intentions in mind, it's just purely a greeting, and particularly where it reflects a cultural practice."
The Australian Muslim Women's Association's Silma Ihram says, regardless of where Muslims stand on the issue, they should shake hands out of respect and courtesy.
"I follow the Shafi'i school (of Islamic law), so I prefer, where possible, not to shake hands. But, I don't want to offend anybody. So the most important priority, in terms of priorities, is that you establish good relations with the people who are around you. And if their custom is to shake hands and they would be offended by not shaking hands, then, for me, it's most important that I, first of all, shake hands and then, once the person is comfortable that I mean no offence, to explain that I prefer not to."
Ms Ihram calls for respect from both sides, but she asks Muslims not to be rigid.
"I would encourage all Muslims not to stand on their fine points of Islamic law when they're living in a country which is not Muslim, but to work on the most important aspects, which is to establish good relations with your neighbours."
The three universities identified are Adelaide's Flinders University, Perth's Curtin University and the University of Western Australia.
