The state Department of Human Services says more than 100,000 people have been enrolled in the private option since sign up began last October.
Coverage on the program, which places uninsured on private health plans paid for with Medicaid funds, started January 1st.
Each year lawmakers need 75% support in both the House and Senate to pass annual budgets. Most lawmakers believe there is not that level of support to pass the DHS budget with the $915 million appropriation for the private option included.
On Thursday, the Joint Budget Committee approved amendments to the program including one by representative Nate Bell, R-Mena, to continue the program but end any promotion or advertising for it.
Representative John Burris, R-Harrison, a key architect in the private option legislation passed in 2013, admits there is considerable uncertainty.
"There's just a lot of questions over the complexity of the program, how far we push, how fast we implement," Burris says. "And just in general when you've got a very contentious issue like this everything is going to be questioned."
Bell voted against the private option last year, but says he will vote for the plan next week which includes his no advertising amendment.
However other Republicans who oppose the program say the amendment doesn't go far enough.
State representative Justin Harris, R-West Fork, says he hopes the vote fails on Tuesday to help set the stage for negotiations to end the private option completely.
"If we're going to deal with this in the 2015 session we're going to have more people who may get thrown off of insurance," Harris says.
The House and Senate are not scheduled to meet Friday and will reconvene on Tuesday after a federal holiday on Monday.
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