A deal would put the combined company in a better position to take on U.S. industry leader AMC Entertainment Holdings Inc, and also give it more scale to fight growing competition from Netflix Inc, Apple Inc and other digital outlets.
The deal value of $23 per Regal share represents a premium of about 12 percent to Regal's closing price on Monday and implies an enterprise value - equity plus debt - of $5.8 billion.
Regal shares have risen 13.6 percent since Reuters first reported in November that Cineworld had approached Regal over a potential deal.
Cineworld said it expected the deal to "strongly" add to earnings in the first full year following completion, currently expected in the first quarter of 2018.
The combined company is expected to deliver pretax benefits of $100 million, as well as additional annual benefits of $50 million, the companies said.
Cineworld said it expected to fund the deal through a rights issue to raise about 1.7 billion pounds ($2.3 billion), with the rest provided by committed debt facilities and existing cash.
Cineworld also expects to be able to maintain its existing dividend policy after the deal closes.