"Following detailed consideration of... All of the issues surrounding a potential disposal of the State's shareholding in Aer Lingus, the Government has decided that it will support IAG's proposal," minister Paschal Donohoe said in a statement.
The sale has been a hot topic for Dublin, with severe opposition within the junior government coalition party, Labour, and from unions who warned about future implications for jobs and connectivity.
To quell opposition IAG made commitments on the future of the airline, promising to maintain the brand and a head office based in Dublin and ensure the airline's lucrative landing slots at London's Heathrow airport would not be sold.
IAG will now have to issue a formal offer while the matter is expected to be discussed in parliament, possibly as soon as Wednesday, with a vote necessary before Dublin can complete the sale.
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IAG will also require the support of Irish airline Ryanair, which is the largest shareholder, with a 29.8 percent holding.
Ryanair spokesman Robin Kiely told AFP its board "will consider any offer on its merits, if and when an offer is made".
IAG said in a statement the acquisition had a "compelling and financial rationale" for the group.
"Aer Lingus would maintain control of its brand and operation while gaining strength as part of a profitable and sustainable airline group in an industry that's consolidating.