Marriott said it remained committed to the deal, which would create the world’s largest hotel chain with top brands including Sheraton, Ritz Carlton, and the Autograph Collection.
Starwood’s shares rose 7.5%, while Marriott, which made a $12.18bn (€10.92bn) offer in November, was up 3%.
A deal with the Anbang-led group, whose offer values Starwood at $12.84bn, may not be easy given increased US scrutiny of Chinese-initiated merger and acquisition over security concerns.
Chinese insurance companies are flush with cash and looking to diversify their portfolios before the country’s ageing population starts claiming on their policies.
US assets are also a good hedge against any future weakness in the yuan.
Anbang, the owner of New York’s Waldorf Astoria, is making a big push into the US hotel industry.
It has also agreed to buy Strategic Hotels & Resorts from Blackstone Group for around $6.5bn, a source told Reuters.
“Anbang’s non-binding offer places Starwood shareholders in the difficult position of choosing between Marriott’s bird-in-a-hand firm commitment and Anbang’s two-in-the-bush offer,” Nomura Securities analyst Harry Curtis wrote in a note.
Marriott may slightly improve the terms of its offer and emerge as the winning bidder, Mr Curtis said.
The $76 per share offer by the Anbang-led consortium, which also includes private equity firms JC Flowers and Primavera Capital, tops Marriott’s offer of $72.08 per share.
Marriott’s offer is now worth about $11bn as Marriott shares have dropped 6.5% since it made the offer.