The House wants the state go forward with Medicaid expansion for the three years the federal government would pay 100 percent of the cost, while a commission develops a state plan in six weeks.
But Senate negotiators said they would be leery of any proposal that does not have lawmakers vote to approve expansion.
"It would be very difficult for the Legislature to move forward without a voting process," said Senate Finance Committee Chair Chuck Morse, R-Salem. "If the Legislature is not involved in voting on this issue, we are not going to get anywhere."
The Senate is expected to offer a counter proposal this morning.
The House proposal would have newly eligible Medicaid recipients become part of the state's Medicaid program or continue with their employers' private insurance plans for the three years. The federal government would pay 100 percent of the cost.
Until Tuesday, the House and Senate have deadlocked over the issue of Medicaid expansion under the Affordable Care Act, with Gov. Maggie Hassan and the House leadership in favor, and the Senate wanting to wait until a study commission reviews the impact on the state.
Without an agreement on expansion, budget writers are not expected to reach a compromise on the state's fiscal 2014-2015 operating budget.
The Senate wants a separate agreement on expansion, and says state health and human services officials and the governor's office have not provided enough information to make a decision.
Senate President Peter Bragdon, R-Milford, proposed establishing a commission to study the impact of expansion before taking action and the Senate included that in its budget.
However, several Republican Senators, led by Sen. Nancy Stiles, R-Hampton, have been working with Hassan on a New Hampshire plan for two weeks.
At Tuesday's budget negotiations, House Finance Committee Chair Mary Jane Wallner, D-Concord, said those negotiations have progressed and Tuesday Bragdon asked to see something in writing.
"Many members of the Senate would like to accept the federal dollars," she said. "We can finalize the details to develop the New Hampshire model in a relatively short period of time not the extended time of the commission the Senate has in House Bill 2."
Wallner said under the House proposal the state would not lose $340 million in federal money a year and many New Hampshire working families would receive health coverage.
After Wallner explained the proposal, Morse called it "a good first step" that addresses concerns raised by both the House and Senate, but warned the Senate's response to the proposal may be a different than what the House proposed.
He urged budget negotiators to continue talking. Negotiators have a noon Thursday deadline.
"I believe this proposal addresses concerns expressed by the Senate and moves the discussion forward," said House Speaker Terie Norelli. "We look forward to hearing their thoughts and engaging in a constructive dialogue about our proposal as we work towards a completed budget before Thursday's deadline."
Under the ACA, the state-federal health insurance plan would be expanded to 138 percent of the federal poverty level, which is about $15,000 for an adult. Expansion is projected to add 58,000 people to the Medicaid rolls and bring in $2.5 billion in federal dollars over seven years to health care providers while costing the state from $27 million to $85 million.
Under the proposal the commission would present its plan to the Joint Legislative Fiscal Committee by Aug. 20, and if the committee does not approve a plan, state health and human services official would submit an expansion plan to the Center of Medicaid Services by Dec. 31.
The House's proposal would use the $200,000 the Senate appropriated for its commission to hire experts.
The commission would begin working seven days after the budget is approved by lawmakers.
The nine-member commission would have three members approved by the governor, three by the House Speaker and three by the Senate President.