He will ask the House of Representatives to note that a growing list of nations, including the Netherlands, Belgium, Norway, Spain, Canada and South Africa, allow same-sex couples to marry.
Both Labor and the coalition oppose gay marriage, saying the union is between a man and a woman.
But there are signs some Labor MPs, especially those from the Left faction, want a free voice on the issue.
Labor Senator Doug Cameron has said it is inevitable that federal laws will be changed to allow gay marriage in Australia.
Mr Bandt will move a motion in the lower house that calls on all parliamentarians to gauge their constituents' views on the issue of same-sex marriage equality.
The Greens already have introduced legislation in the Senate that would clear the way for same-sex marriage.
Independent senator Nick Xenophon said he supported civil unions for same-sex couples.
"And I have been meeting with groups that are advocating for it," he told reporters in Canberra.
Senator Xenophon described his view on the issue as "more small-l liberal" than the major parties.
He said there was need for a debate on the issue.
"At the very least we should very quickly move to civil unions which has been the case in Europe and the UK and a number of other jurisdictions," he said. Read more here....