Last week, Marriott appeared to have outbid Anbang with a $13.6bn offer that gained support from Starwood's board.
In a statement, Starwood's board said the new offer was likely to lead to a "superior proposal".
The board, though, has not changed its recommendation in favour of a merger with Marriott.
Starwood owns several hotel brands, including Sheraton, Westin and St Regis.
Under the terms of its agreement with Marriott, Starwood's board can discuss and provide information to bidders that make superior bids.
But under its latest merger agreement with Marriott, Starwood would have to pay a breakup fee to Marriott of $450m if it decided to accept another offer.
Anbang, whose consortium includes private equity firms JC Flowers & Co and Primavera Capital, has been making a push into the US market over the last several year.
It bought the Waldorf Astoria hotel in New York in 2014.
Starwood shares rose 2% on Monday to $83.75 on the higher Anbang offer.
Chinese insurance firm Anbang has raised its offer for Starwood Hotels to $14bn (£9.8bn), extending a bidding war with Marriott for the group.