Last week, House Republicans passed a continuing resolution that would fund the government through mid-December while stripping money for ObamaCare.
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 Lee (R-Utah), has been pushing Republican colleagues to obstruct the spending bill as a way to generate leverage to end the healthcare law.