NEW YORK (Reuters) - French luxury group LVMH LVMH.PA has persuaded U.S. jewelry chain Tiffany & Co TIF.N to provide it with confidential due diligence after it raised its bid to close to $16 billion, people familiar with the matter said on Wednesday.

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The development represents a key milestone in LVMH’s bid for the iconic U.S. maker of engagement rings. Tiffany earlier this month rebuffed LVMH’s initial $120-per-share all-cash offer, arguing it significantly undervalues it.

Tiffany agreed to provide LVMH with access to its books, after the latter sweetened its bid to close to $130 per share, the sources said. Tiffany plans to continue to negotiate for a better offer, and there is no certainty a deal will be reached, the sources added.

LVMH and Tiffany did not immediately respond to requests for comment.

Tiffany shares jumped 3.6% to $127.75 in after-hours trading in New York on the news on Wednesday.

Jewelry was one of the strongest performing areas of the luxury industry in 2018, according to consultancy Bain & Co, which forecast that comparable sales in the $20 billion global market were set to grow 7% this year.

Tiffany, founded in New York in 1837 and featured in the 1961 movie “Breakfast at Tiffany’s” starring Audrey Hepburn, had struggled with falling annual sales and profit since 2015, before a revenue turnaround in 2017.

Under Chief Executive Alessandro Bogliolo, former head of fashion firm Diesel and a Bulgari alumnus, Tiffany has been building up its e-commerce business, and is trying to court younger shoppers with more affordable pendants and earrings and new designs.

However, LVMH believes Tiffany needs to spend more on reinventing and marketing its brands, and that it can achieve this only as a division of LVMH, according to the sources.

LVMH’s 2011 purchase of Bulgari boosted its smallest and newest business division, jewelry and watches, which also includes Hublot and Tag Heuer watches. Nonetheless, that business accounted for 9% of revenue and 7% of LVMH earnings in 2018, about only a fifth the size of its core fashion and handbag business, home to brands like Christian Dior, Givenchy and Louis Vuitton.

Acquiring Tiffany would increase LVMH’s exposure to the bridal and diamond category, as well as to U.S. luxury shoppers. Should there be a deal, LVMH plans to keep the Bulgari and Tiffany’s brands separate, according to one of the sources.

In recent months, Tiffany and LVMH, as well as their peers Kering PRTP.PA and Richemont CFR.S, have seen a dent in their revenue as a result of turmoil in Hong Kong, where anti-government protests have scared off tourists and battered spending in the Chinese-ruled city.