Senate Republicans are focused on ending ObamaCare's Medicaid expansion as they draft their own version of the House's repeal legislation, Sen. John Thune John Randolph ThuneThe Hill's 12:30 Report - Presented by Facebook - Don't expect a government check anytime soon The Hill's Morning Report - Sponsored by The Air Line Pilots Association - Trump contradicts CDC director on vaccine, masks Senate GOP eyes early exit MORE (R-S.D.) said Tuesday.

In an extended conversation with reporters, Thune said Republicans are working on the design of their ObamaCare repeal bill but realize the window for finding common ground isn't very wide.

"I think that [Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellMcConnell focuses on confirming judicial nominees with COVID-19 talks stalled McConnell accuses Democrats of sowing division by 'downplaying progress' on election security Warren, Schumer introduce plan for next president to cancel ,000 in student debt MORE] will allow a certain amount of discussion, but at some point we’ll have to make decisions and we’ll have to put something out there," Thune said.

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He stressed the vote will be held when they have enough support, and they're not there yet.

Similarly to the House bill, Senators want to eventually repeal the law's optional Medicaid expansion. The only question is how long the post-repeal transition period will be.

"There’s an interest among many of our members having a longer phaseout, a smoother glide path" after repealing the expansion, Thune said. "At some point you’re gonna get back to the original [reimbursement rate], it’s just a question of how quickly that happens."

The House narrowly passed the American Health Care Act earlier this month. Among many other provisions, the bill would effectively freeze enrollment in expansion states by ending the enhanced federal contribution at the end of 2017.

Expansion would be gradually rolled back starting in 2020 when states would have to pay for any newly eligible beneficiaries out of their own pockets.

Thune said the Senate is still likely to change the bill to provide more tax credits to better help low-income people afford health insurance, but acknowledged it's still a "work in progress."

Additionally, Thune expressed support for a last-minute provision in the House bill that would allow states to opt out of certain ObamaCare essential coverage requirements. Critics attacked the provision as not protecting people with pre-existing conditions, but Thune said he liked the flexibility it gives to states.

Thune said senators want to have some sort of reinsurance program or high-risk pool to protect people with pre-existing conditions, but haven't narrowed down their options.

Jessie Hellmann contributed.