CONCORD, N.H. — Democratic and Republican political leaders continued Thursday to work behind the scenes on a possible compromise to expand Medicaid in New Hampshire as they publicly lauded different ways to get the job done.

Democratic Gov. Maggie Hassan and legislative leaders have been trying to negotiate a compromise for weeks, but with nothing decided, the Republican-led Senate and Democratic-led House filed different expansion bills Thursday. An estimated 49,000 poor adults would be eligible for Medicaid under the federal health overhaul beginning next year, but legislative authorization is needed to enroll them.

New Hampshire’s current Medicaid program covers low-income children, parents with non-disabled children under 18, pregnant women, senior citizens and people with disabilities. The expansion would add anyone under age 65 who earns up to 138 percent of federal poverty guidelines, which is about $15,000 for a single adult and $32,500 for a family of four.

The House’s bill would use private insurance for adults who already are on employer-sponsored plans and enroll others into the state’s managed care program. The Senate’s plan is similar the first year, but would shift people from the managed care program onto private insurance in 2015 through the federal insurance marketplace.

Both plans would require federal waivers and would end the expansion if federal financial support fell below promised levels. The Senate would give the government one year to approve its use of the marketplace or the expansion to the adults would end.

The Senate plan also would end automatically at the end of three years when federal funding begins dropping unless the Legislature reauthorized it. If New Hampshire were to expand the program, the federal government would pick up the full cost for the first three years and 90 percent over the long haul. New Hampshire’s health care providers would share in an estimated $2.4 billion over seven years.

The main sticking points to a compromise are the length of time allowed to obtain a waiver before adults are sent to the marketplace, and whether all the adults or just those earning between 100 percent and 138 percent of federal poverty limits use the marketplace, said Senate Republican Leader Jeb Bradley of Wolfeboro.

Senate President Chuck Morse said Senate Republicans, who outnumber Democrats 13-11, do not support the House’s plan, but he remains hopeful a deal can be reached with House Democrats and Hassan before a scheduled Nov. 21 vote on legislation.

“I remain hopeful the politics don’t get in the way,” said Morse, R-Salem.