SINN Féin leader Mary Lou McDonald has apologised to "those who were hurt by the IRA".
The Dublin TD said she would like to meet the Orange Order and would attend Twelfth demonstrations "if I'm invited".
The loyal order has consistently refused to hold official meetings with Sinn Féin.
Orange Order grand secretary Mervyn Gibson told the Belfast Telegraph earlier this year there were "335 reasons" not to engage with the party, a reference to loyal order members killed by republicans during the Troubles.
In an interview with the News Letter, Ms McDonald said: "I am heartily sorry for everybody that got hurt. I am heartily sorry to those who were hurt by the IRA."
When asked if that included members of the order, she said: "Of course, I am sorry that members of the Orange Order were hurt in..."
However, she later said it was not a "reasonable position for the Orange Order to say 'well I'm not going to meet with that person because of everything that happened in the past' because everybody could adopt that position, and then where are we?"
When asked if she would attend the Twelfth demonstrations, she said she would if invited.
She added: "I haven't been invited...I will meet everybody; if I could meet every single person, I would meet every single person individually."
The Sinn Féin president's comments came after Leo Varadkar became the first taoiseach to visit the order's headquarters in east Belfast earlier this month.
Orange grand master Edward Stevenson praised Mr Varadkar for paying his respects at a memorial for the order's murdered members, saying the gesture would be "deeply appreciated by many relatives of the deceased, and the Orange membership as a whole".
