The justices voted 5-4 to grant an emergency appeal from the clinics after a federal appeals court had earlier upheld the new Texas law.
Clinics have argued that the measures will lead to a second major wave of Texas clinic closures in as many years.
The southern US state had 41 abortion clinics in 2012 but only 19 remain.
The new restrictions were due to take effect on 1 July.
They require abortion clinics to meet hospital-grade standards and for doctors at the clinics to have admitting privileges at a local hospital.
Critics say both provisions are designed to shut down abortion providers and will force women to seek abortions at illegal and unlicensed facilities.
Monday's ruling means clinics unable to meet the restrictions will remain open at least until the Supreme Court decides whether it will hear the appeal from clinics.
Several abortion clinics in Texas will remain open after the US Supreme Court decided to put strict new measures against them on hold.