Such clauses bar Simon & Schuster, News Corp. unit HarperCollins, Lagardere SCA's Hachette Livre and Verlagsgruppe Georg von Holtzbrinck, the owner of German company Macmillan, from making deals with rival retailers to sell e-books more cheaply than Apple.
The agreements, which critics say prevent Amazon and other retailers from undercutting Apple's charges, sparked an investigation by the European Commission in December last year.
Pearson Plc's Penguin group, which is also under investigation, did not take part in the offer.
The EU antitrust authority, 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 source said on Tuesday.
Antoine Colombani, spokesman for competition policy at the European Commission, said: "We have launched a market test in September and our investigation is still ongoing."
Amazon declined to comment, while Apple did not respond to an email for comments.
Companies found guilty of breaching EU rules could be fined up to 10 percent of their global sales, which in Apple's case could reach $15.6 billion, based on its 2012 fiscal year.
FROWNING ON ONLINE TRADE CURBS
Antitrust regulators tend to frown on restrictions on online trade and the case is a good example of this policy, said Mark Tricker, a partner at Brussels-based law firm Norton Rose.
"This case shows the online world continues to be a major focus for the Commission. They are looking at lots of different aspects of e-commerce, as this can have such a significant impact on consumers, development and innovation," he said.
"These markets change very quickly and if you don't stamp down on potential infringements of competition rules, you can have significant consequences."
UBS analysts estimate that e-books account for about 30 percent of the U.S. book market and 20 percent of sales in Britain but are minuscule elsewhere. Amazon created demand for e-books when it launched its e-Kindle reader, charging $9.99 for each book.
Apple's agency model let publishers set prices in return for a 30 percent cut to the maker of iPhone and iPad.
The U.S. Department of Justice is also investigating e-book prices. HarperCollins, Simon & Schuster and Hachette recently settled, but Apple, Penguin Group and Macmillan continue to fight the allegations.