The furore over whether or not Donald Trump remains associated with his reality show The Apprentice may be the least of his problems.

As the President-elect vehemently denies a conflict of interest arising from his continued involvement with the show, he continues to be dogged by concerns over his administration’s links to Russian President Vladimir Putin.

The Trump administration is widely expected to name oil executive Rex Tillerson to the influential role of secretary of state. Mr Tillerson is known to have close ties to Mr Putin.

Meanwhile, Mark Burnett, the co-creator of The Apprentice, has confirmed that Mr Trump remains an executive producer of the show’s spin-off Celebrity Apprentice for the coming season.

Considering he will have another job as President of the United States to contend with, questions of priorities have be raised.

The President-elect’s response was swift and angry.

Trump senior advisor Kellyanne Conway defended Mr Trump’s decision to stay on as an executive producer for the show, telling CNN he is conferring with experts about “what he is allowed to do and not do as president of the United States … If [serving as executive producer on ‘Celebrity Apprentice‘] is one of the approved activities, then perhaps he’ll consider staying on”.

Reports by @CNN that I will be working on The Apprentice during my Presidency, even part time, are ridiculous & untrue – FAKE NEWS!

— Donald J. Trump (@realDonaldTrump) December 10, 2016

Mr Trump’s involvement with the NBC show raised concerns over conflict of interest, with the president potentially having an interest in a program aired by a media company that also reports on his presidency.

You’re fired!

Celebrity Apprentice returns to the air with new host Arnold Schwarzenegger.

“Were we so concerned about the hours and hours and hours spent on the golf course of the current president? I mean presidents have a right to do things in their spare time, in their leisure time,” Ms Conway argued.

Mr Trump, who famously used the catchphrase “You’re fired!” as he dismissed competitors from The Apprentice, stepped down from the show last year when he entered the Republican presidential race.

I have NOTHING to do with The Apprentice except for fact that I conceived it with Mark B & have a big stake in it. Will devote ZERO TIME!

— Donald J. Trump (@realDonaldTrump) December 10, 2016

The Russian connection

Perhaps more concerning for Mr Trump are the claims of Russian interference in the US election and the close relation held by Mr Tillerson with Mr Putin.

Mr Tillerson has known Mr Putin for more than two decades and Russia awarded the Exxon Mobil chief executive its Order of Friendship medal in 2013.

US relations with Russia have grown frosty since, most recently with the CIA formally determining the Russians had taken steps intended to help Mr Trump win the November election.

Mr Trump, meanwhile, has mocked the US intelligence agencies for their assessment of Russian involvement, with his White House transition office releasing a statement describing the CIA as “the same people that said Saddam Hussein had weapons of mass destruction”.

The New York Times reports that Mr Tillerson, who has no background in diplomacy outside the energy sector, would face the question of whether to maintain sanctions on Russia imposed during its invasion of Ukraine – penalties he has criticised for slowing Exxon’s investments in that country.

Mr Tillerson has little experience of diplomacy outside the energy sector. Photo: Getty

Mr Tillerson’s connections to Russia would likely come under scrutiny during a Senate confirmation hearing prior to his appointment as secretary of state, with Republican Senator John McCain on Sunday saying his ties to Mr Putin were “a matter of concern to me”.

“I’d have to examine it,” he said on Fox News. “You want to give the President of the United States the benefit of the doubt because the people have spoken. But Vladimir Putin is a thug, bully and a murderer, and anybody else who describes him as anything else is lying.”