Mr Robinson met with representatives from Northern Ireland's Muslim community on Thursday evening to set straight comments he made which appeared to defend controversial evangelical pastor James McConnell, whose sermon about Islam is being investigated by police.
Dr Raied Al-Wazzan, from the Belfast Islamic Centre, told UTV they had a "frank" discussion with Mr Robinson.
"We told the First Minister that we are hurt [by] his comments to the Muslim community and [he] privately apologised and we're satisfied with that," he explained.
"He said 'I am sorry if I hurt any Muslim' and we accept that, and also we invited him to visit the Belfast Islamic Centre and he's kindly agreed to visit us with the deputy First Minister as well.
"We would like to arrange this visit as soon as possible, to work together as one community to show to the rest of the people of Northern Ireland that there is no hate crime here...and to bring the best to Northern Ireland."
I made it very clear that none of my remarks were intended to cause anyone any offence or distress, the last thing I would ever have in my mind would be to cause hurt to anyone and if anyone had interpreted them that way of course, I would apologise.
First Minister Peter Robinson
Following the meeting, Mr Robinson said the meeting was "useful" and "valuable".
"It was an excellent opportunity for me to put in context my views on those matters, to assure them that they were a valued part of our society in Northern Ireland. We recognise the contribution that they had made," Mr Robinson commented.
"I was able to therefore indicate what my own comments were, I was glad to hear that they hold the same view as I do, that [being] the condemnation of terrorism."
Mr Robinson previously insisted his remarks were "misinterpreted and given a meaning that was never intended".
A statement from the First Minister earlier said: "I would never seek to cause any insult to any section of our community.
"I believe in building a peaceful and prosperous Northern Ireland and have always endeavoured to work for the betterment of all the people of Northern Ireland."
My remarks, if they are read properly in the context in which they were given, should cause no offence to anyone, if however, people are offended within the Muslim community, I want to meet them to make it very clear what my intentions and my views are.
First Minister Peter Robinson
Mr Robinson went on to defend Pastor McConnell's "right to freedom of speech".
He added: "I will defend his right just as I defend the right of others to express views with which I disagree. People have the right to express their differing views and indeed the essence of democracy is the ability to do so in a way that is free from fear and intimidation."
Pastor James McConnell sparked controversy when he likened "cells" of Muslims in Britain to the IRA and described Islam as a "heathen" doctrine which had been "spawned in hell".
He made the remarks during a sermon at Whitewell Metropolitan Tabernacle in Belfast on 18 May and police are investigating a "hate crime motive" in relation to the incident.
Pastor McConnell told UTV that he stands by his comments and has rebuffed calls for an apology.
The DUP leader weighed in on Wednesday in defence of the pastor, saying it was "the duty of any Christian preacher to denounce false doctrine".
Mr Robinson, who attends the church in question, added: "He's perfectly entitled to do that - it's an appropriate thing for a minister to do." He said he did not trust Muslims who were involved in "terrorist activities" or those "fully devoted to Sharia law".
Meanwhile the pastor's comments have been condemned by deputy First Minister Martin McGuinness and by representatives of the Muslim community in Northern Ireland.