The group’s policy and campaigns officer Preethi Sundaram said such measures could help to boost diversity in the workplace.
She pointed out that 40 years after the Equal Pay Act came into force, women are still earning on average a fifth less than men.
In some sectors, the gender pay gap is as large as 55 per cent.
Ms Sundaram said the most recent government report to look into the issue stopped short of recommending quotas.
However, she insisted: “All the evidence shows that it is the only sure fire way to ensure more women reach the boardroom.”
She stressed: “In politics, business and public life more generally, decisions which affect us all are being made with too few women in the room.”
In addition to quotas, Ms Sundaram said the Fawcett Society wants to see more firms hiring people expressly to monitor their pay policies.
“Greater awareness of the problem is key to tackling it,” she remarked.