In her first set-piece Dáil performance as Labour leader, Ms Burton urged all sides in the dispute to keep talking.
“If the concerts do get underway successfully, I think we should invite him, perhaps, to complete the world tour in Dublin, probably some time after two years.
“I want to encourage the parties to engage in all of the discussions that are necessary to resolve this issue,” Ms Burton said.
Fianna Fáil’s Timmy Dooley, who told TDs that Mr Brooks might be watching the proceedings live on the web, said the Government had offered a chaotic response to the controversy.
“I dare say Garth Brooks will be somewhat confused by the Tánaiste’s request that he finish up his tour here when she has not outlined a mechanism by which it can happen,” the Clare TD said.
Fianna Fáil’s tourism spokesman said Government inaction had allowed the situation to go from “fiasco to farce”.
Mr Dooley, who was repeatedly heckled from the Government benches, branded the situation a “crisis” which would deprive the Dublin economy of an estimated €50m cash injection and damage the country’s reputation abroad.
Referring to an intervention by the Mexican ambassador into the row, and calls for the White House to become involved, Mr Dooley said: “Why did the Taoiseach sit on his hands for a week when even the Mexicans, and the Moneygall native, Barack Obama, are getting involved, yet neither the Taoiseach nor the Government was prepared to do it?
“It is about upsetting the travel plans of 70,000 inbound tourists, disappointing 400,000 ticket holders who have purchased and paid for tickets. It is about the reputational damage to Ireland as a destination for international entertainment events on a large scale at a time when we are trying to attract the Rugby World Cup in 2023.”
Ms Burton attempted to ridicule the Clare TD’s comments, telling him: “I thought the only people you left out were possibly the Brazilians who might be free to be involved at this point in time.”
Ms Burton insisted that new talks, involving Taoiseach Enda Kenny, had emerged “in the light of facts that have come to attention, particularly in the last few days in relation to the process”.
Independent TD Finian McGrath accused Labour TDs of “intellectual snobbery” because Mr Brooks is a country act. “If that was Leonard Cohen they would be jumping up and down,” Mr McGarth said.
Mr Dooley said Mr Brooks might have watched the rowdy Dáil scenes as it was reported he followed a meeting of Dublin City Council on the web.