OLYMPIA — An agreement that would allow the Legislature to fill a potential $2 billion budget hole, finish work and adjourn for the year was announced Wednesday by Senate leaders.

It would allow some high school students who failed a controversial science assessment test to receive their diplomas and approve a delay on Initiative 1351, the class-size reduction ballot measure approved last November.

“It is the deal we needed, to get done,” Sen. Bruce Dammeier, R-Puyallup, said.

Republican and Senate leaders said their members could vote as early as Thursday afternoon on the four-year delay in I-1351, which already has passed the House, and the proposed two-year delay in the state’s biology assessment test, contained in a new bill filed Wednesday afternoon.

The delay on the biology test, which critics say is not a good assessment of some students’ ability, will be retroactive. That will allow about 2,000 seniors who had failed the test as sophomores, taken a remedial course but still failed a makeup, to get their diplomas if that was the only thing keeping them from graduating. Next year’s seniors will also be allowed to graduate under those conditions.

Parts of I-1351 which require the state to begin lowering the number of students in public schools from fourth grade through high school would be deferred for four years. The state would begin reducing class sizes for kindergarten through third grade, but the other grades would wait. The state’s 2015-17 operating budget, which passed at the end of June to avoid a partial state government shutdown, assumed such a delay, but the bill that spelled out the policy failed in the Senate in an early morning vote last week.

Senate Democrats who support faster implementation of I-1351 had balked at the four-year delay as hurting students in the upper grades, and asked for a vote on a bill passed by the House bill that called for changes to the high school assessment tests as a counterbalance. Senate Republicans said approving a delay to the initiative was part of the overall budget agreement and would only agree to a one-year delay on the test requirement. Discussions went through the night, and culminated with the failed vote to delay the initiative at about 6 a.m.

Nothing has moved through either chamber since. Without that delay to I-1351, the state’s operating budget would eventually be $2 billion out of balance, although budget experts said it didn’t create an immediate crisis.

Senate Democrats said Wednesday the latest agreement is stronger than the offer they rejected last week because it allows for an extra year to find ways to help students pass the biology test and doesn’t move up the requirements for math and English tests by a year, as Senate Republicans proposed.

But Dammeier said while the assessment tests can be improved, they are important for making sure students have the skills for the jobs that are available. The Legislature eventually will need the “political courage” to hold to standards, he said.

The House will have to vote on the biology assessment test bill, but that chamber has been the prime mover in giving students relief from the tests, and passed its legislation on the issue by large margins three times already. It must also pass bills connected to the transportation projects that will be supported by a new gas tax that begins on Aug. 1.

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