The watchdog said it was considering complaints in line with its “usual processes”.
Speaking at the Home Affairs Select Committee, Mr Versi said: “We have seen, unequivocally, failures within specifically the Conservative Party when it comes to Islamophobia – not just small failures [but] fundamental failures in every single way.
“I’m talking about leadership within the party… I’m talking about issues when members of Parliament have done certain things and no action has been taken against them.
“I’m talking about when councillors and representatives of the Conservative Party have said or done things which are inappropriate and unacceptable.”
He said that action was sometimes taken, but sometimes not “until the media have been called”, adding: “And then what happens? A few weeks later they’re quietly let back into the party….These types of things are unacceptable.”
Mr Versi said he was not claiming Islamophobia was not happening in other parties, but he was “focussing on where the specific and large-scale issue is”.
Tory leadership contender Boris Johnson also faced criticism during the committee over his previous comments about women who wear burkas.
Speaking at the committee, Iman Atta, director of anti-Islamophobia charity Tell MAMA, said there had been a rise in reports of abuse from women wearing veils or headscarves after the article.
Addressing those running to become the UK’s next prime minister, she added: “Political leadership comes with responsibility… because people will look at the words that a political leader is using and re-use those words on the ground and in the streets.
“We do ask every single candidate… to be responsible with the language that they use.”