Citing people familiar with the matter, the Journal said that a deal could be announced as soon as Tuesday.
Private equity has long been attracted to the sector's reliable cash flow and growing demand from online shopping, with a spate of takeovers by bigger packaging players further spurring their interest.
RPC, which operates in 34 countries, on Monday had a market capitalisation of 2.95 billion pounds ($3.8 billion).
The company said in September that it was in talks on a possible sale to Apollo and U.S. private equity firm Bain Capital, with the deadlines to make a firm offer extended repeatedly since then.
The FTSE mid-cap company ended talks with Bain in December.