Acosta's nomination came after Puzder, CEO of Carl's Jr. and Hardee's parent CKE Restaurants Holdings Inc., withdrew Wednesday, on the eve of his confirmation, hearing after a number of Republican senators said they would not support him.
CNN and NBC were reporting the that Acosta will be Trump's pick.
"I think he'll be a tremendous secretary of labor", Trump said of Acosta at an afternoon news conference before launching into an extended attack on the "dishonesty" of the media.
However, the response to Acosta's nomination from union groups who had staunchly opposed Puzder was much more measured on Thursday.
Puzder has not provided details about why he chose to withdraw his nomination for labour secretary.
But Wisconsin Restaurant Association President Ed Lump said the $15 wage would shut out many people from entry level jobs.
In a March 1990 episode of "The Oprah Winfrey Show" on domestic abuse in wealthy families, a disguised Fierstein said that the executive "vowed revenge" after she leaked spousal abuse allegations to the public. Sens. Susan Collins of Maine, Johnny Isakson of Georgia, Lisa Murkowski of Alaska, Mike Enzi of Wyoming and Tim Scott of SC - all members of the Senate HELP committee - have declined to state how they plan to vote until after Puzder's hearing.
That uncertainty seemed to trickle down to Republicans, who took umbrage with his immigration stances, which ran against Donald Trump's stated objectives.