Minnesota’s attorney general on Tuesday sued Gov. Greg Abbott of Texas over his refusal to extradite an ICE agent facing assault charges in connection with a shooting during the Trump administration’s immigration crackdown in the state.

The agent, Christian Castro, has been held at a jail in South Texas since May on an arrest warrant issued by the county prosecutor in Minneapolis, where he faces four felony assault charges.

Prosecutors in Minnesota accused Mr. Castro of firing a shot through the front door of a Minneapolis duplex on Jan. 14 after a high-speed chase. The shot wounded a Venezuelan immigrant, Julio C. Sosa-Celis.

U.S. Immigration and Customs Enforcement suspended Mr. Castro upon determining that the agent falsely claimed that he fired his weapon after three assailants brutally assaulted him with a broom and a shovel. Surveillance video that captured the incident contradicted that account.

Mr. Abbott, a Republican, has not responded to an extradition request from his counterpart in Minnesota, Gov. Tim Walz, a Democrat. Under Texas law, a person may be held on an extradition warrant for up to 90 days, which means that if no action is taken soon, Mr. Castro, 52, could be released before the end of the month.

“Governor Abbott is trying to shield Castro from being held accountable for these serious charges by refusing to extradite him to Minnesota,” Keith Ellison, Minnesota’s attorney general, said in a statement. Mr. Abbott’s inaction, he added, “violates federal law and Texas law.”

In an email, Agathe Cirasa, a spokeswoman for Mr. Abbott, declined to comment.

The lawsuit contends that governors have an obligation to turn over a person wanted on a criminal warrant from another state once the person’s identity is confirmed. Minnesota officials said it was exceedingly rare for governors to ignore extradition requests. Mr. Abbott has approved several other Minnesota extradition requests in recent weeks, Mr. Ellison said.

Mr. Ellison and Mary Moriarty, the elected county attorney in Minneapolis, said that officials in Mr. Abbott’s office told them this week that the governor did not intend to act on the request to send Mr. Castro to Minnesota.

Ms. Moriarty’s office believes Mr. Castro could flee to Mexico if he is released, she said. She cited unspecified links she said the agent had to that country.

“We do believe he is a flight risk,” she said during a news conference.

Mr. Abbott has long argued that states should help enforce federal immigration law. Mr. Walz and Mr. Ellison, also a Democrat, have been stridently critical of the Trump administration’s immigration policies.

The lawsuit, filed in federal court in the Southern District of Texas, asks a judge to order Mr. Abbott to sign the extradition order.

If that doesn’t happen immediately, the attorney general’s office is asking the court to block Mr. Castro’s release from custody until the extradition matter is resolved.