Gov. Paul LePage unveiled a bill Tuesday that attempts to fix a problem in the welfare system that can discourage recipients from accepting a raise or taking a second job because they would immediately lose their benefits and face an even bigger financial struggle than when they received public assistance.

The Republican governor has sought big changes to Maine’s welfare programs since he took office in 2011, but most of his efforts have been deeply divi- sive and defeated by Democrats in the Legislature. Fixing the so-called welfare cliff, however, appears to be an issue that both sides are embracing this year, providing hope to supporters that something will get done.

“It’s unusual that we have both parties looking at the same idea, the same concept, so hopefully this can get through in some fashion,” LePage said Tuesday.

A mother with two children on the Temporary Assistance for Needy Families program, for example, loses her benefits once she starts making more than $1,023 a month. That’s well below the poverty level, and she would likely still struggle greatly to make ends meet.

LePage is seeking to ensure that none of recipients’ gross earned income is counted in the benefits calculation for the first two months of employment, so they can maintain state support. After those two months, the amount of assistance they’d receive would gradually decline. The calculation would differ slightly for those who work less than 40 hours a week.

Mary Mayhew, commissioner of the Department of Health and Human Services, called it a “sensible realignment of incentives” in Maine’s welfare system that will move people from “poverty to prosperity.”

The bill would also allow people to qualify for transportation assistance benefits for 18 months, up from 12 months. It would allocate $500,000 for small emergency accounts, which families use to cover for health care or education costs. It’s expected to cost $1.3 million annually plus $500,000 from the federal TANF block grant and would be paid for by savings realized when the state imposed a five-year limit on TANF benefits in 2011.

A similar bill was introduced this year by Democratic Rep. Drew Gattine, cochairman of the Health and Human Services Committee. Gattine’s bill would similarly disregard all of recipients’ earned income for two months. But unlike LePage’s bill, his would also apply to those who work less than 20 hours a week.

A DHHS spokesman said the administration opposes another provision in Gattine’s bill that it says would add about 1,800 two-parent families to the TANF rolls.

Chris Hastedt, public policy director for Maine Equal Justice Partners, which supports Gattine’s bill, said she’s concerned that Le- Page’s effort would hurt those who work fewer than 20 hours a week. She also said that the governor’s bill lacks important pieces of Gattine’s proposal, like ensuring that recipients receive child care assistance retroactive to the date they applied.

But she said there appears to be a willingness on both sides to come to a solution on this issue.

“There is common ground, there’s opportunity and we need to grab that and run with it,” she said.