Speaking in the House of Commons on Monday, Mr Campbell said there was an "anomaly" in law which prevents people in Northern Ireland from applying for British citizenship.
He was debating during the Second Reading of the Northern Ireland (Miscellaneous Provisions) Bill.
He said: "There are thousands of people in Northern Ireland, who were born after 1949 when the Irish Republic left the Commonwealth, who were born in the Irish Republic and have lived for decades in Northern Ireland, who are British.
"They are British by courtesy of their tax-paying ability, British by courtesy of their voting arrangements because they are British voters, they are British residents- but they can't hold a British passport and that is an anomaly that has to be addressed and hopefully will be raised at the appropriate point in the progress of this bill."
He commented that people in Northern Ireland "have a right to claim an Irish identity even though they have never been to the Irish Republic, why can't people born in the Irish Republic but have been British for decades and have lived in the United Kingdom?
"They demand the right to have a British citizenship, and they are currently denied that right, so hopefully we will be able to bring amendments to that effect in the course of this bill."
Under the Good Friday Agreement, residents of Northern Ireland have the right to claim Irish, British or dual citizenship.