Burke, the Archbishop of St. Louis, was asked by The St. Louis Post-Dispatch if he would deny Communion to Giuliani if the former New York mayor approached him for the sacrament.
"If the question is about a Catholic who is publicly espousing positions contrary to the moral law, and I know that person knows it, yes I would," the paper quoted the archbishop as responding.
Burke has said of Giuliani: "I can't imagine that as a Catholic he doesn't know that his stance on the protection of human life is wrong. If someone is publicly sinning, they should not approach to receive Holy Communion."
Asked about it Wednesday while campaigning in New Hampshire, Giuliani said:
"Archbishops have a right to their opinion, you know. There's freedom of religion in this country. There's no established religion, and archbishops have a right to their opinion. Everybody has a right to their opinion."
Burke says that anyone administering Communion is morally obligated to deny it to Catholic politicians who support an abortion-rights position contrary to church teaching.
He is expected to push the nation's bishops to take that stance in a document on political responsibility they will issue to Catholics before the 2008 election.
A number of other Catholic presidential candidates also have abortion-rights stances in apparent conflict with church teaching.
"It is a cause of concern for me and for all bishops to find ourselves in this situation," Burke told the Post-Dispatch.
While it is unlikely Giuliani or any other presidential candidate will present himself to Burke for Communion in the next few months, the archbishop's comments revive an issue that could be a factor for churchgoing voters.
In 2004, Burke said he would deny Communion to Kerry, the Democratic presidential nominee who supports abortion rights. Several other bishops have said politicians should refrain from the sacrament if they oppose the church on such an important issue.
Giuliani, a Republican, sometimes evokes his Catholic upbringing as he campaigns for president, yet he declines to say whether he is a practicing Catholic.
In August, when a voter in Iowa asked if he was a "traditional, practicing Roman Catholic," he said: "My religious affiliation, my religious practices and the degree to which I am a good or not-so-good Catholic, I prefer to leave to the priests."
Last week, Giuliani compared the scrutiny of his personal life marked by three marriages to the biblical story in which Jesus said only someone who was free of all sin should try to stone an adulterous woman.
"I'm guided very, very often about, 'Don't judge others, lest you be judged,'" Giuliani told the Christian Broadcasting Network.
"I have very, very strong views on religion that come about from having wanted to be a priest when I was younger, having studied theology for four years in college," he said.
"So it's a very, very important part of my life," he said. "But I think in a democracy and in a government like ours, my religion is my way of looking at God, and other people have other ways of doing it, and some people don't believe in God. I think that's unfortunate. I think their life would be a lot fuller if they did, but they have that right."
Republicans have been most successful with religious voters President Bush, a Methodist, won the Catholic vote over Kerry, a Catholic, in 2004 but Democratic candidates are fighting back and have spoken frequently about their religious beliefs this year.