HB 87 is scheduled to go into effect on July 1.
Attorneys for groups such as the ACLU and the Georgia Latino Alliance for Human Rights filed a motion for a preliminary injunction Wednesday.
The groups claim the injunction "... is urgently need to prevent this unconstitutional law from causing irreparable injury to plaintiffs and similarly-situated individuals."
The groups filed a lawsuit challenging the constitutionality of the law last week, saying it violates the federal government's power over immigration laws and policy.
But Gov. Nathan Deal's office and the author of the law, state Rep. Matt Ramsey, of Peachtree City, have said they believe the law will withstand those challenges, especially after Arizona's immigration law was held up in the courts.
"We worked very hard and very diligently to make sure the provisions were drafted with that in mind every step of the way, to make sure it was ultimately going to pass muster in the courts," Ramsey said last week.
The law empowers police to check the immigration status of certain suspects, and forces some employers to verify employees' legal status.
Supporters said it is necessary to help the state deal with problems associated with illegal immigration.
Opponents said the law could lead to racial profiling, and has already led to labor shortages in Georgia's agricultural sector.