* Dollar steady after 2009 low vs euro, currency basket
* U.S. stock gains show improvement in risk appetite
* Data from U.S., China and Europe bolster optimism
* RBA, ECB meetings eyed
By Aiko Hayashi
TOKYO, June 2 (Reuters) - The dollar steadied on Tuesday after hitting its lowest this year against the euro and a basket of currencies as reassuring economic data and a rally on Wall Street reduced the safe-haven allure of the greenback.
The U.S. manufacturing sector posted its highest reading since September and construction spending notched its biggest increase in eight months in April bolstering the view that the global economy was on the mend..
Generally upbeat manufacturing reports from the euro zone, Britain and China added to that optimism and fuelled a sell-off in the dollar on Monday.
Commodity-related currencies such as the Australian dollar and the Canadian dollar, considered to be higher risk, retreated from eight-month highs set on Monday against the greenback but still hovered near them as oil prices also jumped.
Traders said however trade was likely to be range-bound ahead of central bank meetings in Australia and Europe.
The Reserve Bank of Australia (RBA) holds a policy meeting on Tuesday and is expected to refrain from cutting interest rates for the second straight month from a record low of 3 percent.
The RBA will announce its decision at 0430 GMT along with a brief statement.
"Trade will likely be slow, at least until the end of the central bank meeting in Australia," said Minoru Shioiri, a senior manager of FX trading at Mitsubishi UFJ Securities.
The European Central Bank and the Bank of England both hold policy-setting meetings on Thursday. Both are expected to hold short-term borrowing costs steady, and the ECB may provide more detail on its plan to purchase assets to stimulate the economy.
"Profit-taking may also emerge in currencies that had gained quite a bit so far, such as the euro, if the central bank meeting in Europe mentions a rate cut or some unorthodox measures," Shioiri said.
The dollar index was steady at 79.133 yen after hitting its lowest since mid-December at 78.586 the previous day.
The euro softened 0.1 percent to $1.4145, having rallied to $1.4246 on trading platform EBS on Monday, its strongest since late December.
Financial markets took in stride General Motors Corp's bankruptcy filing on Monday, the third largest in U.S. history and the largest ever in the U.S. manufacturing sector.
Higher risk currencies dipped from multi-month highs. Sterling edged down 0.1 percent to $1.6415, off a seven-month high it hit on Monday at $1.6497.
The Aussie dollar fell 0.4 percent to $0.8065, retreating from an eight-month high of $0.8154, as did the New Zealand dollar which eased to $0.6465.
The dollar also fell 0.3 percent to trade at 96.30 yen. (Editing by Edwina Gibbs)