The U.S. Fish and Wildlife Service (USFWS) released this Tweet on Thursday morning.
Minnesota U.S. Representative Betty McCollum has also requested the U.S. Attorney's Office look into whether Palmer has violated any U.S. laws, she wrote on her Facebook page.
Although he issued a statement via a PR representative and wrote a letter to his patients since it emerged Tuesday he was responsible for the beloved lion's death, CNN reports the doctor's whereabouts are unknown, saying he has gone "underground."
It appears as though federal authorities have not been able to track him down yet either, given the public appeal from the USFWS.
The 55-year-old, whose River Bluffs Dental office has been closed since becoming the subject of threats and Wednesday protests from angry citizens, is an avid bow-and-arrow hunter who paid more than $50,000 to kill the lion on July 6.
However, it was later revealed that Cecil the lion had been illegally lured by Palmer's hunting guide to private land from his home in the Hwange National Park, where he was the showpiece attraction for visitors because of his distinctive black mane, and was protected from hunting.
The 13-year-old had been collared as part of a conservation research project being conducted by Oxford University, with the collar cut off before the lion was decapitated and skinned.
In his statement, Palmer said he "deeply regrets" his role in the animal's killing, adding that he believed he was taking part in a legal hunt.
Authorities in Zimbabwe have also said they want to talk with Palmer as part of their investigation into Cecil's death, which saw his hunting guide Theo Bronkhorst appear before a court on poaching charges Wednesday.
Could Palmer face prosecution in the U.S.?
In terms of what action could be taken against him in the United States, Slate says African lions aren't endangered or threatened under U.S. law, so it was not illegal to kill the animal as per the Endangered Species Act.
However, the website argues that he could face action if the $55,000 he paid the wildlife guides for the privilege of killing Cecil is considered to be a bribe, which The Independent suggested Tuesday.
In this case, Slate notes, he could face action in America under federal laws that forbid "foreign travel with the intent to engage in certain 'unlawful activities' overseas," which includes bribery.
Bloomington police meanwhile are investigating "terroristic threats" made against Palmer, who has been the subject of a barrage of online abuse this week, and told Reuters it would be "investigated like any other similar offense report."
Although patrols have been stepped up around Palmer's business and his home in Eden Prairie, local police have said they are not providing him with personal protection, according to this Eden Prairie police statement.