By Harry Robertson and Tom Westbrook

LONDON/SINGAPORE (Reuters) -The yen rallied against the dollar for a fourth straight session on Tuesday as investors positioned for the possibility that the Bank of Japan will tighten monetary policy next year while the Federal Reserve loosens.

The dollar hit its lowest level since mid-September at 147.16 yen and was last down 0.61% at 147.45.

More broadly, the dollar index, a gauge of the greenback against six other currencies, fell to its lowest since late August at 103.17 and was last 0.13% weaker at 103.32.

"There has been a lot of excitement, momentum is building, about the ability of the Bank of Japan to exit its ultra-loose monetary policy... possibly next year, ending negative interest rates," said Jane Foley, head of FX strategy at Rabobank.

Foley said a sharp drop in the dollar was also encouraging investors to unwind some of their bets against the yen. "The dollar is weaker, and this I think is just the catalyst for the market making bets on how far dollar-yen can really move," she said.

U.S. yields have tumbled as investors have wagered that the Federal Reserve will cut interest rates next year, after a slowdown in U.S. inflation in October.

That has dragged the dollar index down from an almost one-year high at the start of October, when U.S. economic data was consistently beating expectations.

The 10-year U.S. Treasury yield fell for a fourth session running on Tuesday to 4.39%, after falling on Monday in the wake of a solid auction of 20-year bonds. It hit a 16-year high above 5% in October.

The euro rose to its highest since mid-August at $1.0966 on Tuesday and was last very slightly higher at $1.0944.

Sterling was up 0.2% at $1.253, after hitting a two-month high of $1.254. Bank of England Governor Andrew Bailey on Monday said it's "far too early to be thinking about rate cuts" in Britain.

Also weighing on the U.S. currency was a rally in China's yuan, which hit an almost four-month high of 7.13 per dollar.

China's central bank set the midpoint of the yuan's trading band at its strongest since Aug. 7. The firm currency fixing came together with a Bloomberg News report on forthcoming support for the property sector which boosted stocks, said National Australia Bank strategist Rodrigo Catril in Sydney.

Elisabet Kopelman, U.S. economist at lender SEB, said: "Strong risk appetite and speculation about future interest rate cuts are not a good environment for the dollar, which has come under pressure against a number of Asian currencies this morning."

Minutes from the Fed's last meeting are due at 1900 GMT and headline the day ahead, along with a speech from European Central Bank President Christine Lagarde.

Some analysts caution that the dollar's downward momentum may not have too much further to run. "There is a risk that we are going to get push-back about the pace of Fed easing," said Foley.

(Reporting by Harry Robertson in London and Tom Westbrook in Singapore; Editing by Jacqueline Wong and Susan Fenton)