"The price action in the currency market both yesterday and today, with the greenback generally on the defensive...suggest the positive implications of the tax proposal have been broadly anticipated," said Nick Bennenbroek, head of currency strategy at Wells Fargo Securities in New York.
"In essence, and as we have often seen around US Federal Reserve rate hikes, we appear to be seeing a repeat of US dollar 'buying on the rumor and selling on the fact'," he added.
The US Senate approved the tax bill early Wednesday morning, but had to send it back to the House of Representatives, which had passed it on Tuesday, for another vote due to a procedural foul-up. The outcome is not expected to change.
The bill essentially slashes taxes for corporations and the wealthy, while offering mixed, temporary tax relief to working individuals and families.
Some analysts were skeptical about the tax bill's supposed positive impact on the US economy.
While increased fiscal stimulus should support the dollar as it would allow the Fed to raise interest rates at a faster pace, it also could expand fiscal deficits with little incentive for businesses to hire more workers, said Omer Esiner, chief market analyst at Commonwealth FX in Washington.
"It remains unclear how much fuel this bill will provide to the economy at this point in the business cycle," he added.
In mid-morning trading, the dollar index fell 0.2 percent to 93.295. That led the euro to rise 0.3 percent versus the dollar to $1.1872
Year-end portfolio moves and a rise in German bond yields also helped the euro, analysts said.
The dollar also fell against sterling, the Canadian and New Zealand dollars.
Against the yen, however, the dollar rose to one-week highs and was last 113.16 yen, up 0.2 percent.
Gains in the dollar were limited versus the yen, as many market players looked to Thursday's outcome of the Bank of Japan's two-day policy meeting for clues to whether it will join the US Federal Reserve and European central banks in winding back stimulus.
The Swedish crown, meanwhile, rose on Wednesday against both the euro and the dollar after the country's central bank called an end to its quantitative easing policy.