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.

Each week in the UK, about 12 seemingly healthy people aged 35 or under are victims of sudden cardiac death, largely due to undiagnosed heart conditions.
If you have a family member who has been diagnosed with an inherited heart condition, or has a family history of premature deaths, doctors will suggest you be tested.
Having angina or a heart attack at a young age can also prompt a test.
Young people should also be checked if they have:
They carry out an electrocardiogram test - also known as an ECG.
This looks at the electrical activity of the heart.
It is painless and involves having small stickers placed on the chest, which are connected to the ECG machine.
A cardiologist checks the results.
Sometimes, an echocardiogram - an ultrasound test that looks at the structure of the heart - is also done.
If someone under 35 has died suddenly, the NHS will offer "cascade testing" to family members to see if they have inherited a faulty gene.
Just being identified helps. People can then be monitored, and encouraged to make any lifestyle changes that could help them.
There are drug treatments available - and people can be fitted with internal cardioverter defibrillators - a matchbox-sized device placed just under the collar bone that gives the heart electric pulses or shocks to get the heart rhythm back to normal.
Other procedures and surgery are also available.
Talk to your GP if you have any concerns.
These websites have more information:
Chinese insurance firm Anbang has raised its offer for Starwood Hotels to $14bn (£9.8bn), extending a bidding war with Marriott for the group.
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About 620,000 people in the UK have a faulty gene that puts them at risk of developing coronary heart disease or sudden death, and most are unaware, heart experts have warned.