Speaking to reporters at a separate event at the White House earlier on Thursday, Trump said he was nominating a man to the post but gave no other details.
Potential candidates included former National Labor Relations Board members Peter Kirsanow and R. Alexander Acosta, according to a Bloomberg report.
Joseph Guzman, an assistant professor at Michigan State University, and Catherine Templeton, former head of the South Carolina labor department, were also potential candidates, Bloomberg reported, citing a White House official.
The official told Bloomberg that Trump had met with Acosta and Guzman Wednesday night and had earlier planned to meet with Templeton on Friday.
Reuters could not immediately confirm the report.
Puzder, the chief executive officer of CKE Restaurants Inc., removed his name from consideration on Wednesday amid concerns that he could not garner enough Senate votes to be confirmed.
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