"The board of directors unanimously decided to issue a favourable opinion of GE's offer" and will begin consultations with personnel, an Alstom statement said.
French law requires the company to consult workers’ representatives and gain regulatory approval relating to foreign investment rules.
GE boss Jeff Immelt hailed the deal as "good for France, GE and Alstom" but pointed out that it would not be finalised before next year.
His counterpart at Alstom, Patrick Kron, called it a win-win-win situation, good for Alstom employees as well as GE and the French state.
The French government, concerned about a foreign concern owning a company that provides material for nuclear power plants and builds the TGV high-speed train, intends to buy a 20 per cent stake in the company from the Bouygues conglomerate.
But Bouygues, which currently holds 30 per cent of Alstom shares, wanted 35 euros a share, while the government was offering the current price of 28 euros.
After a long day of negotiation on Saturday, sources told the AFP news agency that there was agreement in principle and talks restarting late on Sunday morning were likely to wrap up the deal.
Economy Minister Arnaud Montebourg had said that the GE-Alstom sale could not go ahead if Bouygues was not prepared to sell the shares.