"People are increasingly thinking that the payroll tomorrow will probably also be decent given the state of the data we had today," said Jens Nordvig, global head of currency strategy at Nomura in New York.
He said a strong report would raise the likelihood of the Federal Reserve raising rates in September. Fed rate hikes are expected to boost the dollar by driving investment flows into the United States.
Nordvig said greater risk appetite led traders to bet against, or "short," the euro, which hurt the currency.
Read MoreRobertson: Euro could go below parity
In addition, Prime Minister Alexis Tsipras urged Greeks to reject an international bailout deal, wrecking any prospect of repairing broken relations with EU partners before a referendum on Sunday that may decide Greece's future in Europe.
The television address came less than 24 hours after Tsipras wrote a conciliatory letter to creditors asking for a new bailout that would accept many of their terms. The address put additional pressure on the euro, analysts said.
"I still think the market generally interprets news about a prospective Greek exit as news that's negative for the euro," said Thierry Albert Wizman, global interest rates and currencies strategist at Macquarie in New York.
The euro was last down 0.97 percent against the dollar, at $1.10440. The dollar was last up 0.56 percent against the yen, at 123.180 yen.
Against the Swiss franc, the dollar was up 1.46 percent, at 0.94890 franc, after hitting its highest level in three and a half weeks, at 0.94900 franc.
The dollar index, which measures the greenback against a basket of six major currencies, was last up 0.87 percent at 96.319.