Richard Braine, who was recently elected UKIP's sixth leader in the last three years, suggested the Islamic holy book should have its distribution limited.
He said: "Don't forget that our law actually bans incitement to violence and so I think it's a very important issue that people look at - why the Manchester Arena attack, Lee Rigby, 7/7 bombings, Westminster Bridge, London Bridge and many, many other attacks - Asah Shah, for example, in Glasgow."
Asked whether he believed the distribution of the Koran in public should be outlawed, Mr Braine added: "We need to look into whether it's illegal to do so because we do have laws against incitement to violence, so we should look into that.
"I understand there's something like a 100-year threshold where, if a work is more than 100 years old, it isn't subject to incitement laws.
"But I think that's worth looking at because there's no question that some people who commit violence are motivated by the scripture."
Mr Braine stressed he was "not saying" the Koran should be banned completely, but drew a comparison between Islam and the Scientology movement.
He said: "It's important to analyse why people behave the way they do - if religions or cults are actually encouraging people to do bad things then we should be able to talk about that and look at it carefully and actually see if there's a solution to stop people just answering the demands or the commands to go and be violent."
UKIP finished last in this month's Brecon and Radnorshire by-election, with the party gaining fewer votes than the Monster Raving Loony Party.
Mr Braine admitted: "UKIP needs to fight its way back up in the polls again, there's no question about that.
"The primary reason why UKIP has done badly, electorally, since 2016 is because we actually forced the other parties to deliver our policy - Brexit.
"And we forced them to say: 'Yes, that's what we're going to put in our manifestos'."
Mr Braine is reported to have made similar comments about the Koran at a recent hustings event for UKIP members.
According to the Guardian, the software developer and photographer also claimed that British Islam has particular problems with bigamy and welfare abuse, and referred to the jailed anti-Islam activist Tommy Robinson as a "political prisoner".
The newspaper also reported Mr Braine branded Islam an "alien ideology" and suggested there were areas of the UK where "you're simply not welcome" as a non-Islamic person.
A UKIP spokesman told Sky News it was neither the party nor Mr Braine's policy to ban the distribution of the Koran in public.
A spokesman for the Muslim Council of Britain said: "UKIP's continued descent into even greater depths of bigotry is only matched by its growing irrelevance and obscurity.
"Virulent Islamophobia was not a successful tactic under the previous UKIP leadership, and will not be this time. Our nation knows better."
Mr Braine replaced his political ally Gerard Batten as UKIP leader, whose term in charge ended shortly after the EU elections in which the party lost all its MEPs.
Mr Batten attempted to stand as a candidate in the subsequent leadership contest but was barred by UKIP's ruling body from seeking re-election.
As well as UKIP's poor electoral performance, Mr Batten's spell as leader was also marked by a mass exodus of leading figures following his decision to employ Robinson (real name Stephen Yaxley-Lennon) as an adviser.
Mr Batten, who supported Mr Braine's candidacy, has himself described Islam as a "death cult".
Former UKIP leader Nigel Farage left the party in December last year due to its "fixation" with Islam under Mr Batten.
He went on to form the Brexit Party, which topped this year's EU elections.
List of UKIP leaders since EU referendum in 2016:
Nigel Farage - resigned September, 2016
Diane James - resigned October 2016
Paul Nuttall - resigned June 2017
Henry Bolton - removed by vote of no confidence February 2017
Gerard Batten - term ended June 2019, barred from standing for re-election
Richard Braine - elected August 2019