The conservative legal campaign against the centerpiece of Mr. Obama’s immigration overhaul has largely succeeded in running out the clock, blocking the president’s executive actions from taking effect while judges consider their legality. Now, even if Mr. Obama ultimately prevails in the legal battle — which would occur next summer at the earliest — there will probably be time for at most a few hundred thousand of those immigrants to qualify for protection before the end of the president’s term.

Worse for the administration, in the next few weeks, the states fighting to stop Mr. Obama may score their biggest victory yet — achieving a long-enough delay in the lower courts to prevent the Supreme Court from even considering and ruling on the case until after next year’s presidential election. That timing would leave any final decision about immigration to Mr. Obama’s successor.

The immediate holdup is the United States Court of Appeals for the Fifth Circuit, in New Orleans, where a three-judge panel has been deliberating the president’s actions for more than three months — far longer than the court’s goal to decide “within 60 days after argument” in most cases. White House officials are bracing for a loss in the appeals court, in part because two of the three judges have already ruled against the administration in an earlier decision.

But Mr. Obama needs a ruling soon from that panel so his lawyers can try to persuade the Supreme Court to take the case. If the decision does not come quickly, his hopes for a late-hour appeal to the Supreme Court this year will disappear.