Sky News reports the buyout firm – an arm of Bain Capital – is expected to provide a cash injection in the region of £30m to parent company Wheel Topco as part of the refinancing.
City sources told Sky the deal has yet to be finalised and that the figures were subject to further negotiations.
Pizza Express operates c.360 restaurants across the UK, with a further 110 company-owned and franchise sites in international markets.
The group plans to reach 1,000 restaurants globally by 2030.
Last month, it restructured its central restaurant support team, which led to around 50 redundancies.
A spokesperson said at the time: “The difficult decision was made to ensure we can address current economic conditions.”
In 2021, parent company Wheel Topco secured £335m senior secured five-year bonds at 6.75%, maturing in July 2026.
The business reported losses of £350m in 2019, as it struggled with a £1.1bn debt pile, under former owner Hony Capital, and was then forced to enter a CVA during the pandemic, leading to the closure of around 15% of its then 450-strong estate.