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8 year oldAccording to a former producer of the show, Mr Trump is heard making offensive remarks in the recordings.
But Mark Burnett, the London-born president of MGM Television, said he "does not have the ability nor the right" to release the material.
Mr Trump appeared in the US show from its inception in 2004 until last year.
Mr Burnett created The Apprentice and also produced it, along with a range of other reality TV programmes.
At the weekend, recordings of the Republican presidential candidate making obscene remarks about women in 2005 were released by The Washington Post and NBC.
Mr Trump is heard commenting to Access Hollywood presenter Billy Bush that "you can do anything" to women "when you're a star" and brags about trying to grope and kiss women.
He did not realise his microphone was on at the time, and has apologised for making the comments.
After the recordings were released, Bill Pruitt, who worked on the first two seasons of the US version of The Apprentice, tweeted: "As a producer on seasons one and two I assure you: when it comes to the Trump tapes there are far worse."
Following Mr Pruitt's comments, several media outlets began speculating about the contents of the footage and whether it could be made public.
One producer claimed she was asked to sign an agreement that the recordings would not be leaked, adding there would be a financial penalty if they were.
But on Monday evening Mr Burnett and MGM Television issued a joint statement, denying that legal action was threatened.
Joint statement from Mark Burnett and MGM:
"MGM owns Mark Burnett's production company and The Apprentice is one of its properties.
"Despite reports to the contrary, Mark Burnett does not have the ability nor the right to release footage or other material from The Apprentice. Various contractual and legal requirements also restrict MGM's ability to release such material.
"The recent claims that Mark Burnett has threatened anyone with litigation if they were to leak such material are completely and unequivocally false.
"To be clear, as previously reported in the press, which Mark Burnett has confirmed, he has consistently supported Democratic campaigns."
Mr Burnett was raised in Essex before emigrating to the US in 1982, when he was 22.
He went on to work for MGM recordings and became known as a significant figure in reality television.
In addition to creating and producing The Apprentice, Burnett created formats such as Survivor, The Voice and Shark Tank - the US version of Dragon's Den.
He became president of MGM Television in December 2015.
Last month, he attended the Emmy Awards in Los Angeles, where he accepted the prize for outstanding reality competition for The Voice.
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