The companies said they had reached agreement over a recommended cash and share offer.
The offer represents a premium of about 30% to BPI's closing price as of 8 June, RPC said in a statement.
Shares in BPI soared by about a third after the proposed transaction was announced.
The deal has still to receive regulatory clearance.
RPC is a multi-national company which specialises in rigid plastics.
BPI produces about 275,000 tonnes of polythene films a year for a wide range of markets, including agriculture and horticulture and industrial and consumer products.
It is also one of the largest recyclers of waste polythene film in Europe.
Northamptonshire-based RPC said the deal was "strategically compelling", providing RPC with an established platform in the European polythene films market.
BPI chairman Cameron McLatchie said: "The last five years have seen consistent improvements in BPI's performance and prospects, but not all of this progress has been reflected in the price or rating of BPI's shares.
"RPC has recognised the value inherent in our business and prospects by making an offer at an attractive premium to the share price.
"BPI's business should benefit from the ability of a larger group to expand its footprint in Europe and beyond.
"Shareholders will benefit from enhanced liquidity for their investment, and employees will have access to the opportunities available in a larger group.
"The BPI board is therefore unanimously recommending acceptance of the offer."
RPC chief executive Pim Vervaat said: "The proposed offer for BPI represents a compelling strategic opportunity for RPC to enter the European polythene films market through an established platform. BPI has a strong product portfolio with attractive market positions in its core markets in Europe.
"The combination will further broaden RPC's range of polymer conversion technologies in line with global peers, establish a new growth platform with a strong cost synergy potential whilst enhancing the group's overall polymer-buying capability.
"I believe the combination of RPC and BPI is an excellent strategic fit and look forward to growing the enlarged platform to continue generating value for our customers and our shareholders."
Plastics packaging group RPC is set to buy British Polythene Industries (BPI) in a deal which values the Greenock-based company at £261m.