The Apprentice host was on the ITV show to talk about the BBC programme's semi-final tonight.
However talk soon turned to President Donald Trump and his former role as the host of the US edition of The Apprentice.
Winding Lord Sugar up, Piers asked who out of the two businessman could write the bigger cheque.
Lord Sugar, 71, then replied: "I could write a cheque using my total funds around £250,000,000 and present it to the bank and it would be paid.
"That t****r with the hair, he couldn't do that."
Piers said quickly: "That's no way to talk about the President of the United States and I'd like to apologise to anyone offended by that."
Earlier in the conversation Lord Sugar said he was really pleased with the current run of The Apprentice and he had signed up to do more.
He said: "Excellent series, it's good, it keeps going, the 14th series.
"They've hired me for another two series because, you know, they just need me."
It's the second time this week that Piers, 53, has been forced to apologise on air.
On Monday's show, the host said sorry after he exclaimed "bloody hell" after seeing a flying Santa.
Piers was left gobsmacked as he interviewed Bear Grylls, while a man in a Father Christmas suit did indoor skydiving him behind him.
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