The oil services firm said its all-share offer of 564p-per share represented a 15% premium on Amec's share price at the close of business last Friday.
The agreement, which will give Wood Group 44% of the new company, was unanimously recommended by both boards but remains subject to a shareholder vote.
Amec Foster's share price closed 12% higher on the FTSE 250.
Wood Group's stock was 1% higher. Both have felt pressure from the collapse in world oil prices since 2014.
Loss-making Amec's decision to focus on the oil sector ultimately left it vulnerable to takeover. It said a £500m rights issue it had planned, aimed at slashing its debt pile, was being suspended as a result of the offer.
The pair said the deal would result in savings in the region of £110m per year.
Wood Group chairman Ian Marchant said: "The combination extends the scale and scope of our services, deepens our existing customer relationships, facilitates further development of our technology-enabled solutions and broadens our end market, geographic and customer exposure.
"Delivering significant sustainable synergies will also result in a leaner and more competitive combined group, creating value for shareholders."
If the deal is approved, Mr Marchant will continue as chairman of the new company while Wood Group's chief executive, Robin Watson would also remain.