The language from Tester and Corker would amend a bill aimed at increasing funding for the Public Works and Economic Development Administration. The Senate agreed Tuesday to take up this bill after Sen. Jim DeMint (R-S.C.) dropped his objection.

DeMint was concerned that Senate Majority Leader Harry ReidHarry ReidTop Lobbyists 2017: Grass roots Boehner confronted Reid after criticism from Senate floor GOP in uncharted territory rolling back rules through resolutions MORE (D-Nev.) would not allow an open amendment process on the bill, and threatened Monday night to block it from reaching the floor, a GOP aide told The Hill on Tuesday.

Once DeMint received assurances  from the majority on Tuesday that an open process would be followed, he agreed to let the bill come to the floor without further delay.

The aide noted, however, that if Reid does not keep his promise, DeMint expects the Senate GOP caucus to block further progress on the bill.

DeMint's procedural allowance for the bill is not an indicator of support for the underlying legislation, which would increase funding for the EDA to $500 million through 2015, compared to just under $300 million in 2011, according to his office.

“This bill is nothing more than another failed stimulus idea that takes money from workers and businesses and gives it to Washington to pick winners and losers," DeMint told The Hill in a statement on Tuesday night.

"We’ve already wasted hundreds of billions of tax dollars on a misguided stimulus that left us with record high unemployment, and we don’t need to repeat the mistake."

The Senate adjourned at 5:55 p.m. on Tuesday and is set to reconvene at 9:30 a.m. on Wednesday.