Doug Richard was questioned by police and released on bail.
A spokesman for City of London Police said: “City of London Police detectives arrested a 56-year-old man on January 5 2015 on suspicion of inciting a child to engage in sexual activity, sexual activity with a child, meeting a child following sexual grooming and rape of a girl under the age of consent.
“He was bailed on January 6 until March 2015.”
The alleged victim is said to be 13 years old.
In a statement to a Sunday newspaper. Mr Richard said: “I absolutely deny the allegations made about me. It would be inappropriate to comment further given that the police are at a very early stage of their investigation.”
Mr Richard, who is from California, has launched and sold a number of successful technology and software businesses.
He starred in the first two series of Dragons’ Den, the BBC business programme.
He has produced reports on business support and apprenticeships for the Conservative Party in opposition and the Government.
He launched the School for Startups in 2008 and followed it up with the School for Creative Startups three years ago. Both have operated courses in Sheffield.
Mr Richard received an honorary Queen’s Award for Enterprise Promotion for his work promoting, developing and helping entrepreneurs in 2006.
In 2010, he received the Enterprise Educator of the Year award from the National Council on Graduate Employment for his work through School for Startups with UK universities.
Mr Richard is a serial entrepeneur but has said he loves teaching too.