GMB's backing means the offer has been accepted by four unions representing National Health Service (NHS) workers whose members include midwives, physiotherapists and ambulance workers.
The NHS Staff Council, which includes representatives from NHS employers and trade unions, is due to meet on May 2 to vote on whether to accept the offer.
The GMB's leadership said it would now vote in favour of the pay offer, after 56% of its members who voted in a ballot accepted the deal.
"This new pay offer would not have happened without the strike action taken by ambulance and other GMB health workers," said Rachel Harrison, GMB National Secretary.
"Our members recognise that progress has been made - from the government originally offering nothing, health workers will be thousands of pounds better off."