Apollo has offered 782p per share for RPC, together with a previously declared interim dividend of 8.1p.
RPC said the offer represents a 15.6% premium to its closing price of 683.6p on September 7, the last business day before announcing takeover talks.
Ahead of the deal announcement, the deadline for Apollo to make a firm offer or walk away had been extended several times since the two started talks.
Bain Capital was also in talks to make a bid for RPC but dropped out of the race last month.
Offer price below analysts' expectations
“It’s taken months to get to this stage, and the final offer price is some way behind what analysts had suggested was possible," said Nicholas Hyett, equity analyst at Hargreaves Lansdown.
"It’s not impossible for a rival bidder to emerge, but plastic packaging is far from popular at the moment and with the global economy looking rocky, it might be asking a bit much to expect a rival to fork out the best part of £4bn to keep RPC out of private equity hands.
"It’s potentially disappointing for investors – a 15.6% premium is hardly over generous and the shares have traded higher in the last 12 months. But with board backing and minimal regulatory concerns, we’d expect the deal to go through.”
RPC says offer recognises the quality of its businesses
RPC said it intends to recommend shareholders vote in favour of the offer from Apollo.
"The board believes that the offer recognises the quality of RPC's businesses and the strength of their future prospects," said RPC chairman Jamie Pike.
"In July I stated that differing investor views on the appropriate level of gearing were constraining the group's ability to pursue opportunities for growth and, as such, putting pressure on RPC's valuation.
"I also said that the board was working to resolve this situation.
"Today's announcement is the culmination of that process."
The news comes as RPC faces slowing demand amid a regulatory clampdown on plastic waste in the EU.
In mid-morning trading, shares rose 4.65 to 767p.