With President-elect Donald Trump's campaign rhetoric on illegal immigration still fresh on their minds, legislative Democrats&nbsp;have readied a pair of proposals they believe&nbsp;will offer some&nbsp;immigrants additional legal help.The bills, set to be introduced on the first day of the new legislative session Monday, primarily aim to bolster the legal representation of immigrants who are in the country&nbsp;illegally and threatened with deportation. California&nbsp;has no formal role in national immigration policy, but the bills could supercharge the state's role in pushing back against a Trump administration's effort to deport as many as 3 million people living in the United States.Most sweeping is a bill that would authorize state government grants to nonprofit organizations that provide legal help for immigrants facing deportation. Dubbed "due process for all" in a summary document obtained by The Times, Democrats believe the money could help a significant number of immigrants to successfully challenge deportations.The second bill would establish a new training and funding program for public defenders involved in immigration cases.Democrats in the Legislature have strongly pushed efforts in recent years to help those who are in California illegally, and legislative leaders made it clear last month that they intend to fight against the ideas floated by the president-elect during the campaign.The bills will also present an early test of the Democrats' new supermajority status in the state Capitol. Both will be drafted with an "urgency clause," which requires a supermajority vote in both houses and would allow the laws&nbsp;to take effect immediately if signed by Gov. Jerry Brown.