Apollo and Evri expect the deal to complete later this year. The announcement comes after weeks of speculation over the future of Evri.
Advent International, another American buyout fund, put the company up for sale earlier this year. The auction has attracted a series of suitors including rival delivery company DHL and the Chinese internet giant JD.com.
While neither side revealed financial terms of the deal, it is thought to have been worth £2.7bn. Advent had paid around €1bn (£840m) for the majority stake in Evri and a minority stake in German group Otto in 2020.
Martijn de Lange, the Evri chief executive, said Advent had “provided the business with a strong foundation for continued expansion”.