SYDNEY, Oct 21 (Reuters) - Cabcharge Australia Ltd, the country's only listed taxi payment company, said it had rebuffed an approach from FleetCor Technologies Inc to buy its core payments business.

Australian media reported that the New York-listed firm had made an approach to acquire the Cabcharge unit for A$500 million ($439 million).

Sydney-based Cabcharge said on Tuesday that FleetCor had made "an initial approach to investigate a transaction", adding that the FleetCor approach had no financial terms.

"Having considering the options available to Cabcharge, the board chose not to pursue FleetCor's approach," Cabcharge said in the statement to the Australian Securities Exchange.

Australian inbound M&A activity fell the most in a quarter of a century in the three months to end-September as plummeting commodity prices and a faltering local stock market challenged the appetite of overseas investors for the country's assets.

But for the year to Sept. 30, inbound M&A activity grew 39 percent to $32.4 billion, the highest since 2011, with most inbound deals originating in the United States.

Cabcharge shares have lost 22 percent since Sept. 15 after the government of New South Wales state cut the additional fee that it can charge customers on top of their taxi fare.

The shares were up 2 percent in early trading on Tuesday, valuing Cabcharge at A$558 million at market prices. (1 US dollar = 1.1390 Australian dollar) (Reporting by Byron Kaye; Editing by Ryan Woo)