Neither Mr. Peyerk nor Mr. Jackson responded to phone calls and emails seeking comment on Sunday.
For more than three decades, federal regulators did not issue import permits for black rhinos until the Obama administration issued three starting in 2013 as their population rebounded in Africa with stricter conservation management.
The Trump administration issued two — in 2017 and 2018 — and Mr. Peyerk’s, if approved, would be its third.
Though President Trump in a 2017 tweet called trophy hunting a “horror show,” his administration has reversed Obama-era restrictions on the import of endangered elephant and lion trophies from several African countries. Donald Trump Jr. and Eric Trump, the president’s sons, have hunted big game and photos from a 2012 hunting trip drew criticism.
The Fish and Wildlife Service notified the Humane Society of the United States last week that it plans to approve Mr. Peyerk’s permit application, Kitty Block, the organization’s president and chief executive, said.
That news was met with criticism from animal rights and conservation groups.
“The Trump administration has dealt another blow to wildlife protection,” Sara Amundson, the president of the Humane Society Legislative Fund, said in a statement.
The number of black rhinos may have increased in recent years but poaching has also increased, said Ms. Block, who is also the chief executive of Humane Society International. Before 2014, there were no reports of black rhino poaching but within two years, there were 80, she said.
“With rhino poaching on the rise, now is the time to ensure that every living black rhino remains safe in the wild,” Ms. Block said in a statement. “Black rhinos must be off limits to trophy hunters.”