IAG has made a third bid for Ireland's national airline the BBC reports which will value the company at $1.6 billion.
The Aer Lingus board discussed the offer on Friday and apparently agreed to the deal.
The CEO of IAG is Irish-born Willie Walsh, a former CEO of Aer Lingus.
The Irish government owns 25 percent of Aer Lingus and if the board accepts the bid the government will have to make a decision to pass Ireland’s national airline out of Irish control.
The other major owner is Ryanair with 29 percent. Their own bid to take over a majority of Aer Lingus was rejected on lack of competition grounds.
A key element of the deal is Aer Lingus' take-off and landing slots at Heathrow Airport which are extremely valuable. Each pair of slots is estimated to be worth $45 million dollars.
IAG was created in 2009 after the merger of British Airways and Iberia.Fianna Fáil transport spokesman Timmy Dooley has already called on the Irish government to block the sale, saying the landing slots were was being “dramatically underestimated” and a sale would be damaging for Dublin, Cork and Shannon Airports.
He claimed any sale of the airline could see its critical Heathrow slots “siphoned off, risking future connectivity and jobs at Dublin, Cork and Shannon airports,” he said.
The fear is that the new airline owners will ignore smaller markets in Ireland.
A senior Irish government source told the BBC: "The issue of landing slots at Heathrow airport is crucial for the Irish government and was discussed at a parliamentary party meeting on Wednesday.
"You have to have flights from Dublin and Cork into a major hub and the Irish government would drive a hard bargain."