The man was questioned by North Yorkshire Police on Friday.
He has been released on bail pending further inquiries, a force spokeswoman said.
Last week, Mr Karim, Conservative MEP for the North West of England, told the BBC that he had received threats.
The politician said: “Other than the type of conduct that I have had to face, there has been very direct threats made, which are now currently with the police, are being investigated and I anticipate that there are individuals who will be spoken to as a result of those threats.”
A North Yorkshire Police spokeswoman said: “A 55-year-old Harrogate man was arrested on Friday 17 June 2016 in connection with social media messages sent to an MEP.
“He has been released on bail pending further inquiries.”