The Bureau of Land Management has been seeking public feedback on an operation targeting the area known as the Calico complex, which covers five designated herd management areas.
The muster had been scheduled to start early in December. The bureau says the start date has been put back to December 28.
Court action to have the muster stopped has been taken by the animal welfare group, In Defense of Animals (IDA) and ecologist Craig Downer.
The IDA and Downer were to this week file a motion for a permanent injunction, with supporting affidavits from horse experts and eyewitnesses to Bureau of Land Management (BLM) roundups.
The motion ask Judge Paul Friedman, of the US District Court for the District of Columbia, to stop the roundup altogether.
The Equine Welfare Alliance welcomed the delay, adding that the bureau has so far received more than 10,000 public comments on the proposed round-up.
President John Holland said: "We view this action as a positive step toward a full moratorium on wild horse and burro round-ups that would provide an opportunity for a much needed assessment of the situation.
"We are profoundly grateful to IDA and the law firm of Buchanan, Ingersoll and Rooney for taking on this essential action on a pro bono basis."