Congress must continue working to fix a “broken” immigration system, and Thursday’s preliminary Senate compromise on the issue may require tweaking, U.S. Sen. John Cornyn, R-Texas, said Thursday from Washington in a conference call with reporters

“We have borders that are insecure and an immigration system that is literally unenforceable,” said Cornyn, the top Republican on the Senate Judiciary Committee’s Immigration, Border Security and Refugees subcommittee.

“Congress has a clear responsibility to repair that broken border. Simply wishing the problem will go away will not make it so.”

Document fraud and identity theft help make the current immigration status unenforceable, he said. He also called for doubling the size of Border Patrol and boosting worksite enforcement, among other measures.

“I do not support a repetition of the amnesty that passed in 1986,” Cornyn said. “Granting legal status to 12 million people is problematic.”

Any immigration deal that is not enforceable is “rotten to the core,” he said. “I do think the president is eager to sign an immigration bill. I also believe it’s the Senate’s and the Congress’ responsibility to send him a good bill.”

Cornyn said while the broad principles of the compromise may be laudable, there is still work to be done.

“As far as I’m concerned this is not a final bill but a work in progress,” he said.

Senators may not see exact language of the bill until Friday, Cornyn said. “I’m reserving judgment until I see what the specific legislation looks like,” he said, adding that he expects the full Senate to take up the debate on Monday.