Last week, House Republicans passed a continuing resolution that would fund the government through mid-December while stripping money for ObamaCare.

Senate Majority Leader Harry ReidHarry Mason ReidMcConnell not yet ready to change rules for Trump nominees The Hill's Morning Report — Sponsored by CVS Health — Trump’s love-hate relationship with the Senate Trump to press GOP on changing Senate rules MORE (D-Nev.) is expected to advance H.J.Res. 59 but add an amendment that removes the language that defunds ObamaCare. A vote to end debate on the motion to proceed is expected Wednesday even if Cruz continues to speak on the Senate floor.

The government will shut down on Oct. 1 if Congress doesn't approve some form of government funding measure.

Cruz has criticized the healthcare law because he said it would harm the economy and has mandates requiring people to but health insurance. He said President Obama has issued unfair exemptions for big businesses and predicted the same would happen for unions.

Obama delayed an employer mandate until 2015 that require businesses with more than 50 full-time employees to provide health insurance to employees. Unions have also asked for an exception to the law so that members can keep their existing healthcare plans and employer contributions, while still qualifying for tax rebates to help cover insurance costs for low-income workers.

Cruz, along with Sen. Mike LeeMichael (Mike) Shumway LeeSenate GOP urges Trump administration to work closely with Congress on NAFTA Senate panel advances Trump's CIA nominee Doug Jones to oppose Haspel as CIA chief MORE (R-Utah), has been pushing Republican colleagues to obstruct the spending bill as a way to generate leverage to end the healthcare law.