That’s a much tougher approach than the one being sought by Cruz’s fellow Texans — GOP Sen. John Cornyn and Democratic Rep. Henry Cuellar — who would leave the directive alone. Instead, they would toughen a 2008 human trafficking law while speeding up immigration proceedings and authorizing 40 more judges to handle the cases. The Obama administration has expressed openness to revisiting the 2008 law, and the Texans’ plan could be included in a spending measure that will soon be considered by the GOP-controlled House.

The move is vintage Cruz: stake out a staunchly conservative position on the biggest debates in Congress, whether it’s pushing to defund Obamacare at the expense of a government shutdown or now trying to end a controversial Obama policy aimed at deferring deportation proceedings for certain undocumented immigrants who came to the United States as children.

While plenty of Republicans would be happy to end the Obama directive, some think it’s unrealistic to tie it to an assistance package. And there’s no sign that the House, dominated by Republicans, is considering such a move.

Supporters of the effort say targeting the directive would resolve the root cause of the current crisis, but some fear that a fight over the directive could simply delay getting aid to the border.