At the time of writing women occupy more than 25% of board positions in the UK’s top 100 companies. Davies will be extending this target to 33% by 2020 and to cover FTSE 350 firms too.
Lord Davies told BBC Radio 4's Today programme: "The focus of our work was on fixing the boardroom. We've done that and there's very few all-male boards left.
"There's been a cultural change in the UK corporate scene, which was totally dominated by men. That is not the case today. We now need to see the same change, through a voluntary approach, in the executive committee structure of big companies and small companies."
Krystyna Nowak, managing director of board practice at search firm Norman Broadbent, warned that while encouraging progress has been made businesses could continue to manipulate statistics by appointing women to non-executive positions.
“With the FTSE 100 reaching its 25% women on boards objective ahead of target, we expect this new goal of one-third of women on FTSE 350 boards to be met before the end of this decade,” she said.
“However, this is a missed opportunity as the recommendations in the report are somewhat disappointing. By only increasing the target for the percentage of women on the Plc board many companies will just continue the trend of appointing women to non-executive director positions, which manipulates board composition to achieve targets and leaves the executive board to continue to be dominated by men.
“The onus now lies on the CEOs of our country's top Plcs to promote females to C-level positions and lead the reform of a gender equal executive pipeline. The government now has a chance to address these issues in its equality boosting measures planned for 2016."
The Lord Davies review coincides with Labour MP Chuka Umunna and Liberal Democrat former business secretary Vince Cable calling for an end to all-white boards. In an article for the Evening Standard, they said: “If gender diversity has increased, the ethnic diversity of our boards has gone into reverse.
“We cannot carry on like this. It sends a terrible message to young Britons of ethnic minority heritage. It fails to make the most of the family connections and cultural understanding of emerging markets that diaspora communities can bring.”
In the Women on Boards Davies Review Annual Report 2015, released in March, Davies described the progress of increasing numbers of women on boards as “little short of a revolution".
“I have never doubted that Britain has extraordinary talent,” he said. “Nor that there are plenty of credible, experienced women, willing and capable of serving on British boards. Business needs to unleash the full extent of this talent. It is good for business, good for the UK economy, and ensures our future competitive position in the global arena.”