The firms have agreed to sell a range of assets to ease the path of the deal.
The move comes after the Competition and Markets Authority found the merger could curb competition in the supply of services to the UK oil industry.
It told the firms to offer a remedy or face an in-depth investigation.
On Tuesday, the CMA said it was considering new undertakings put forward by the companies.
The proposals would see Amec Foster Wheeler sell off "almost all" assets which contribute to engineering, construction and maintenance services in its upstream offshore oil and gas business in the UK.
The watchdog added that there were "reasonable grounds" for believing these undertakings, or a modified version of them, might be acceptable.
It will now open a public consultation on the proposals.
The CMA's decision-maker in the case, Kate Collyer, said: "It is crucial that competition is maintained in this major UK industry.
"We will consider the undertakings offered by Wood Group and Amec Foster Wheeler further, and carefully consult interested parties, in order to make sure that they fully address our concerns."
The CMA's final decision on the takeover is due by 12 October.
Wood Group chairman Robin Watson described the CMA's announcement as an "important milestone", adding that it gave the firm "further confidence in our ability to complete the transaction in quarter four this year".
Amec Foster Wheeler also welcomed the latest development.
Chief executive Jon Lewis said: "Our offer of a proposed remedy in May and the early commencement of the sale process to potential buyers of the upstream oil and gas business has ensured we have navigated this process ahead of schedule, increasing the likelihood that the transaction with Wood Group will close in quarter four this year."
If it goes ahead, the deal would create a company with a combined value of about Â£5bn.
Regulators are considering proposals by energy services firms Wood Group and Amec Foster Wheeler to address competition concerns over their planned Â£2.2bn merger.