The men had covered the license plate of their pickup with duct tape, BLM officials have said.
Non-law-enforcement employees in the West Desert district of the agency are removing BLM logos from their vehicles as a precaution, said Eric Reid, the wrangler's supervisor at the U.S. Bureau of Land Management's Fillmore Field Office.
However the BLM's state director, Juan Palma, said Thursday that he remains calm despite the threat on the wrangler.
"We always are worried about our employees, but I believe that most citizens here in Utah are very law abiding," Palma said.
Utah Highway Patrol Sgt. Todd Royce said Thursday that troopers were investigating the threat but had no new leads.
Anti-BLM sentiments are on the upswing as the agency contends with controversies on multiple fronts:
Last month, militants joined Nevada rancher Cliven Bundy in a standoff against federal agents over Bundy's unpaid grazing fees.
In the past two weeks, ranchers in western Utah have threatened to round up wild horses against federal law, saying the herd has grown beyond the BLM's own management plan and is crowding out grazing cattle. State and county officials have supported the ranchers.
And, on Saturday, ATV riders, led by a San Juan County commissioner, plan to protest BLM restrictions by driving through southeast Utah's Recapture Canyon, where motor vehicles are forbidden.
Those events are intensifying long-standing tensions between some Utahns and the federal government, which owns two-thirds of the state's land.
The Associated Press contributed to this story.