It is understood that detectives are preparing a file to be sent to the Crown Prosecution Service for charging advice, but it is likely to be several weeks before the file is submitted.
The force said in a statement: “On Tuesday, 14 July a man in his 20s was arrested by Met officers on suspicion of sending threatening communications to a Member of Parliament.
“The arrest relates to a social media post from earlier this year, which was reported to police on Friday, 8 May.
“After receiving the report, detectives submitted an application to a social media platform to gain access to the user’s contact information.
“After the relevant information was returned to detectives, the man was arrested, with support from local Met officers, at a residential address in south London. After being held in police custody overnight, he has since been bailed pending further enquiries.”
Mr Farage has previously disclosed that his home was the subject of a firebomb attack last year, which has not so far resulted in any arrests.
On Wednesday, Zia Yusuf, the home affairs spokesman for Reform, spoke of the “overwhelming sense of fear amongst the Reform MPs” following Miss Widdecombe’s killing, which prompted police to offer security to all of the party’s MPs.
He said: “In terms of the current situation with state protection for Reform MPs, there has been some movement in the very near past. I’m not going to comment on that but suffice to say I think, particularly in light of new information, I’m pleased to say the threats are being taken seriously.”