The case underscores the battle between retailers and publishers over pricing control of e-books, which publishers hope will boost revenue and increase customers.
Apple and the publishers have offered to let retailers set their own prices or discounts for a period of two years, and also to suspend "most-favoured nation" contracts for five years, the sources said.
Such clauses bar publishers from making deals with rival retailers to sell e-books more cheaply than Apple.
The EU antitrust authority was investigating the companies' e-book pricing, which critics say prevents Amazon and other retailers from undercutting Apple's charges.
Pearson Plc's Penguin group, which is also under investigation, did not take part in the offer.
The European Commission, which in September asked for feedback from rivals and consumers about the proposal, has not asked for more concessions, said one of sources.
"The Commission is likely to accept the offer and announce its decision next month," the person said on Tuesday.