Speaking during a five day trip to Lebanon, Jordan and Israel, the Fine Gael Minister spoke of his outrage at the attacks.
The racist and offensive messages were also posted online after they were distributed.
The Justice Minister warned those responsible that incitement to hatred is a criminal offence under Irish law and confirmed he has already raised the issue with police.
The Minister told the Irish Independent: “I utterly condemn racism and religious bigotry in all of their forms and am appalled by the nature of these communications.
“Religious intolerance has no place in our society. Incitement to hatred and incitement to violence are offenses under our laws.”
Ireland’s Immigrant Council has called on anyone who received the vile messages to contact police.
The report says the hate mail letter, written in block capitals, threatened to attack any Muslim man, woman or child at any mosque in Ireland.
It also outlined opposition to the building of a large, new mosque on the north side of Dublin.
The letter said: “Muslims have no right to be in Ireland. The Irish people are not happy with your presence in our country which belongs to the true Irish people.
“This land belongs to the Christian faith and we will not allow you to turn it into a Muslim country.
Just remember there are more of us and we have more guns than you will ever see. A mosque and Muslims are the devil and a legitimate target and we will attack.”
A spokesman for the Islamic Cultural Centre in Dublin said the Muslim community was not worried by the threat.
Dr Ali Saleem, a key figure in the center in Clonskeagh, Dublin told the Irish Independent: “The letter was written by immature people. It shows ignorance of the reality.
“The person who wrote that failed to understand that most of the Muslims in Ireland, 65,000 of them, are Irish. They are living at home.
“There are 49 mosques in Ireland, nine in Dublin, that again is a highlight that people living in Ireland, regardless of their beliefs and race, they enjoy the freedoms of beliefs and practice. That’s very much something that’s in harmony with Irish heritage.”