According to the Irish Times, government sources say the group will make further concessions on the airline's Heathrow access in an attempt to get the deal over the line.
Representatives from IAG met with Government officials yesterday and are due to meet again next week.
It's understood IAG may tackle concerns about jobs and connectivity, in order to convince the coalition to sell the State's 25 percent share in Aer Lingus.
Meanwhile Aer Lingus passenger traffic dropped slightly last month.
Ten thousand more passengers were carried on long haul routes, at 56 thousand.
But short haul traffic decreased by almost three percent.
The airline says it was due to planned network adjustments as a result of seasonal demand.
The total number of passengers carried by Aer Lingus last month was 627,000.
