Crossbench senators will push ahead with their crusade to amend Australia's racial discrimination laws, foreshadowing plans to "split the Liberals" in the hope of forcing the Turnbull government's support.

Their persistence comes despite conceding the moves are doomed to fail as the numbers currently stand, with the Coalition unwilling to revisit the idea and the crucial Nick Xenophon Team resolutely opposed.

Two private members' bills will come before the Senate: one, from Liberal Democrat senator David Leyonhjelm, to entirely abolish section 18c of the Racial Discrimination Act, and another from Bob Day, to remove the words "offend" and "insult" from the controversial clause.

Either bill would be supported by newly elected Victorian senator Derryn Hinch, who senses an opportunity to divide and (possibly) conquer.