(Reuters) - Alphabet's Google proposed on Friday a loosening of its agreements with Apple ( AAPL ) and others to set Google as the default search engine on new devices to address a U.S. ruling that it unlawfully dominates online search.

The proposal is much narrower than the government's bid to make Google sell its Chrome browser in the antitrust case over online search.

Google urged U.S. District Judge Amit Mehta in Washington to move cautiously in deciding what the company must do to restore competition after his ruling that the company holds an illegal monopoly in online search and related advertising.

While Google plans to appeal that ruling at the end of the case, it says the upcoming "remedies" phase should focus on its distribution agreements with browser developers, mobile device manufacturers, and wireless carriers.

The judge found the agreements give Google a "major, largely unseen advantage over its rivals" and result in most devices in the U.S. coming pre-loaded with Google's search engine.

The agreements are hard to exit, the judge said, especially for Android manufacturers, which must agree to install Google search in order to include Google's Play Store on their devices.

To fix that, Google could make them non-exclusive and, for Android phone manufacturers, unbundle its Play Store from Chrome and search, the company said in court papers.

Unlike the government's proposal, Google's would not end revenue sharing agreements, which pass a portion of ad revenue Google makes off of search users to the device and software companies that present it as the default search engine.

Independent browser developers including Mozilla, which makes Firefox, have said the funds are crucial to their operations. Apple ( AAPL ) received an estimated $20 billion from its agreement with Google in 2022 alone.

Google's proposal sets the stage for a trial Mehta will hold in April, where the U.S. Department of Justice and a coalition of states will seek to show the need for wide-ranging remedies, including making Google sell off Chrome and potentially its Android mobile operating system.

Prosecutors also want Google to stop paying to be the default search engine, and cease investments in search rivals and query-based artificial intelligence products, and license its search results and technology to rivals.

Prosecutors say the proposals aim to spur innovation in online search, where Mehta found Google's overwhelming market share keeps competitors from gathering the search data needed to improve their products. Prosecutors also seek to prevent Google from extending its dominance in search to AI.

(Reporting by Jody Godoy in New York; Editing by Muralikumar Anantharaman)