* Dollar index just shy of July 20 high
* Australian dollar slips as central bank holds rates (New throughout, updates prices and market activity to U.S. market open; new byline, changes dateline, previously LONDON)
NEW YORK, Nov 7 (Reuters) - The dollar rebounded against a basket of currencies on Tuesday, as investors renewed bets that monetary policy will continue to diverge between the United States and the euro zone.
The dollar index, which tracks the greenback against six major currencies, was up 0.35 percent at 95.091. The index is just shy of the three-month high of 95.150 hit late last month.
The euro was 0.31 percent lower to $1.1573 against the dollar. Its session low was $1.1555, its lowest since July 20.
"There is a renewed focus on the diverging policy outlook between the Fed and most other major central banks," said Omer Esiner, chief market analyst at Commonwealth Foreign Exchange in Washington.
"The idea that the U.S. is going to normalize policy more quickly than other major central banks was largely put on the back burner for most of this year," he said.
"That story has once again come back into the spotlight as a result of the ECB meeting most recently," he said.
Late last month, the European Central Bank prolonged its bond purchases well into 2018, diminishing chances it would raise interest rates next year. Meanwhile, investors expect the U.S. Federal Reserve to raise interest rates next month followed by roughly two more hikes next year.
Esiner also said the dollar drew support from "cautious optimism" that lawmakers in Washington were making progress toward an eventual deal on taxes.
Republican lawmakers on Monday began revising their proposed overhaul of the U.S. tax code, as Democrats pointed to the loss of popular deductions as proof the legislation was an assault on the middle class.
"That may be a maybe a very marginal driver of the dollar strength. If there are signs of progress that could quickly become a bigger force," said Esiner.
Sterling slipped against the dollar, giving up some of its gains from Monday, with Britain set to enter the next stage of negotiations on its departure from the European Union on Thursday. The British pound was 0.43 percent lower against the dollar.
Elsewhere, Australia's central bank on Tuesday left its cash rate at a record-low 1.5 percent, and it looked likely to remain sidelined for months, with inflation low and debt-laden consumers cautious.
The Australian dollar slipped 0.68 percent to $0.7638 against the greenback. (Reporting by Saqib Iqbal Ahmed; Editing by David Gregorio)