New crossbench Senator David Leyonhjelm has flagged the possibility of cutting a deal with the Federal Government on asylum seekers in exchange for support for gay marriage.

Senator Leyonhjelm will introduce legislation into the Parliament later this year aimed at legalising same-sex marriage.

The Liberal Democrat says gay marriage has become an "emblematic" issue for the left, but insists it is a "liberty" issue and not just a "left wing" one.

The last federal parliament rejected several attempts to allow gay marriage in Australia – Labor allowed its MPs a conscience vote on the issue, but the Coalition did not.

Senator Leyonhjelm has called on "closet libertarians" in the federal Liberal Party to vote according to their conscience, and not party policy.

He said he would be willing to support temporary protection visas for asylum seekers who come to Australia by boat, in exchange for Government MPs being granted a conscience vote.

"The temporary protection visa issue is neither here nor there to me, so if the Government wants my support there is going to have to be a contra," he told reporters at Parliament House.

He said "true liberals" believed the Government should stay out of people's private lives.

"It is not the job of the Government to define relationships," he said.

"In fact the day will come when we look back on the role of governments to grant marriage certificates based on gender as the bizarre misappropriation of power that it is."

Senator Leyonhjelm referenced Labor's Penny Wong and Prime Minister Tony Abbott's sister Christine Forster as he made his argument.

"Today I challenge Labor, Mr Shorten and Labor Party members to look Senator Wong in the eye and tell her she does not deserve equal rights," he said.

"Today I challenge Mr Abbott and Liberal Party members to do the same with Mr Abbott's sister."

New Senator may also push to legalise euthanasia

Senator Leyonhjelm says he is also considering a push to legalise euthanasia as he is very much in favour of assisted suicide.

He plans to introduce legislation if the Greens are unsuccessful.

"Richard Di Natale, the Greens Senator, has a committee looking into that so I'm going to give that all the time it needs to continue through to completion," he said.

"That is an issue important to us as well, and so if he fails then I will look at whether I can do it and succeed."