* Dollar index gains almost 3 percent since start of month

* Greenback near 2-month high after Yellen fans rate hopes

* Australian dollar jumps on strong data

By Jemima Kelly

LONDON, May 31 (Reuters) - The dollar climbed towards a two-month high against a basket of currencies on Tuesday and was on track to record its strongest month in six on growing expectations U.S. interest rates will rise in the coming months.

Against the yen, the greenback was set to post its best month in 1-1/2-years , having gained around 4.5 percent since the start of May after a number of intervention warnings from Japanese officials pushed down the Asian currency.

The latest trigger for dollar strength came from Federal Reserve Chair Janet Yellen, who on Friday said a rate increase in the coming months "would be appropriate" if the economy and labour market continued to improve.

The dollar index, which measures the greenback against a basket of six major peers, rose 0.3 percent on Tuesday to 95.802 , close to the previous day's two-month high of 95.968 and putting the dollar on track for its best month since November.

In London, Societe Generale macro strategist Kit Juckes said he did not expect the Fed to raise rates in either June or July, but that the fundamentals - namely economic and policy divergence between the United States and other major economies - pointed towards further dollar strength.

"The question for me here is whether the dollar can carry on rallying on the prospect of the Fed raising rates faster over the next 18 months than is priced in, as opposed to rallying only on expectations of a move in June or July," he said.

An exception to the greenback's broad strength on Tuesday was the Australian dollar, which jumped almost 1 percent on better-than-expected domestic economic data. Still, the Aussie is on track for an almost 5 percent fall this month - its worst performance in almost a year.

Against the yen, the dollar rose towards a one-month high, trading up 0.1 percent on the day at 111.16 yen.

The dollar also fetched a one-month high of 111.455 yenon Monday before stepping back to around 111.09 yen, almost 6 yen higher than a 1-1/2-year low struck earlier in the month against the Japanese currency.

If U.S. payrolls due on Friday show solid job growth and if Yellen signals a rate hike in her speech on the following Monday, the dollar could break above the previous April peaks close to 112 yen, said Koichi Takamatsu, manager of forex at Nomura Securities.

But that scenario may not materialise if the spectre of a U.S. rate hike hurts broader risk sentiment and boosts demand for safe havens such as the yen, Takamatsu added.

(Additional reporting by Hideyuki Sano in Tokyo; Editing by Tom Heneghan) ((jemima.kelly@thomsonreuters.com; +44)(0)(20 7542 7508; Reuters Messaging: jemima.kelly.thomsonreuters@reuters.net))