Chinese insurer Anbang is set to acquire hotel operator Starwood Hotels & Resorts Worldwide for about $13.2 billion, the firm's biggest push yet into the global real estate market.

Anbang's approach to Starwood first became public earlier this week, and Starwood said Friday it had determined Anbang's offer was superior to a deal struck last year with Marriott.

The Chinese firm, whose offer for Starwood is already fully financed, has been an aggressive buyer of U.S. hotel properties of late. It already owns New York's iconic Waldorf Astoria. The current $78-per-share cash offer includes J.C. Flowers & Co. and Primavera Capital as partners.

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Starwood's original suitor, Marriott, now has five days to respond. It struck a $12.2 billion deal to buy Starwood in November. Mariott said in a statement that it continues to believe it is a better merger partner and that it is considering its options.

Sources told CNBC that Marriott is expected to make a counter-bid.

With a bidding war on the cards, shares quickly passed the Anbang offer price. Starwood's stock rose 4.7 percent to $80 in early trading.