He said such a series "would show what a great service the programme is performing for young entrepreneurs, young people, to inspire them".
The BBC said there are no current plans to broadcast a spin-off series.
Speaking about some of the show's past winners, Lord Sugar said: "They've turned into great business people."
He added: "I think the BBC need to do something about that. They need to do a programme, for example, on where they are now."
Earlier this year, BBC Two broadcast a spin-off programme from Dragons' Den, Pitches to Riches, which charted the trajectory of several business ideas the so-called Dragons had invested in.
Lord Sugar added that the success of previous Apprentice winners was "what motivates me to carry on doing [the show]".
"They're doing very well, and they don't get enough publicity. The BBC doesn't get enough publicity over the purpose of this programme.
"I enjoy watching these young people grow a business from scratch really, and that's why I get a great buzz out of it."
"Ricky Martin [2012 series winner] has got about 25 people working for him, Tom [Pellereau, 2011 winner] has eight or nine working for him, Leah [Totton, 2013 winner] has got two clinics now."
He added: "They are one of the reasons why the candidates flock to the auditions because they want to be them."
Not every previous contestant on the show has gone on to success in the business world, however.
One of the most high-profile former candidates is Katie Hopkins - who is now a newspaper columnist and radio presenter, well-known for her controversial opinions
"You get a few people that are picked up by the media because of their outlandish comments, and they go and pursue a separate career," Lord Sugar said.
"Lots of them have tried. Katie Hopkins has found herself a special niche in the market. She's a bit like Piers Morgan on steroids, and of course the media lap it up. And the more controversial she wants to be, the more they lap it up."
Claude Littner and Karren Brady have returned to assist Lord Sugar on the new series of The Apprentice, which begins on 6 October on BBC One.
The businessman said the show's tasks had been updated for 2016.
"One of the things I love about this process is, as I'm getting older, I've got a great production team that keep me on my toes as far as technology is concerned.
"With a lot of the tasks some of the stuff we're going to be doing didn't exist 12 years ago. For example, there's a virtual reality episode in there."
Lord Sugar said he believes "the freshness of the contestants" keeps the show relevant, adding: "One of them was 11 years old when The Apprentice first started and has watched it since then and here they are, turning up to be one of the contestants in the show."
"We have a new audience growing every year, and that's the excitement of the programme."
Lord Sugar was speaking hours after Republican presidential candidate Donald Trump took part in a televised debate with the Democratic candidate Hillary Clinton.
From 2004 until last year, Trump appeared as the figurehead of the US version of The Apprentice - the position Lord Sugar holds on the UK show.
Asked for his thoughts on his former US counterpart, Lord Sugar said: "Well he's not as good as me, as far as The Apprentice is concerned. He's not in my class, quite frankly.
"I saw the debate with him and Hillary Clinton and what comes to mind is that it's like a US version of Would I Lie To You?
"As many people have said it is quite a frightening thought that he may be the most powerful man in the world shortly."
Karren Brady added: "For me personally, I think Hillary Clinton, another woman leader across the world, means the world is a safer and better place."
Lord Sugar also did not shy away from talking about UK politics, commenting that the outcome of the EU Referendum in June made him "fear for our country".
Referring to the vote to leave the EU, Lord Sugar said: "I was very clear in my views, I think they've made a fatal mistake."
"Lots of people are saying nothing has changed. Well, exactly. Nothing has changed because we haven't even started [the process of leaving] yet."
Lord Sugar, previously known as Sir Alan Sugar, became a peer in 2009, while Karren Brady became Baroness Brady in 2014.
So would Claude Littner like to be in line for a knighthood of his own in the future?
"To be honest it crosses my mind all the time," he joked, "but no-one has actually made me the offer".
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Lord Sugar has said he would like the BBC to commission a new programme focusing on the progress of past winners of The Apprentice.