Apollo saw off competition on the deal from rivals such as Chinese online retailer JD.com and DHL, according to reporting by Sky News.
The sale of the Bradford-based delivery company, which delivers over 720 million parcels a year, is expected to be completed in the first quarter of 2024, according to Advent.
Since acquiring Evri in 2020, Advent said it has invested almost £200m into the business and tripled its size.
Evri chief executive Martijn de Lange said: “The last five years have seen transformative change in our proposition which has allowed us to take full advantage of the positive trends in our markets.
“With the support of Advent, the business today is in a strong position and momentum is encouraging. We are very excited to be partnering with Apollo and optimistic for what the future holds.”
Formerly known as Hermes, Evri drafted in advisers earlier this year to explore its options, including a potential sale, after it emerged that numerous businesses were eyeing up a bid.