The dollar's strength against the euro was a "spillover" from last week's commentary from the Fed, said Alan Ruskin, global head of currency strategy at Deutsche Bank in New York.
German 10-year Bund yields held near a record low of 0.14 percent touched last week, also hurting demand for the euro.
The dollar weakened against the Japanese yen, meanwhile, after hitting a three-week high of 120.845 yen earlier in the session. Analysts cited comments on Monday from Koichi Hamada, an economic adviser to Japan's Prime Minister Shinzo Abe, indicating that the yen was excessively weak against the dollar.
The Bank of Japan's massive monetary easing aimed at boosting growth and lifting inflation to a 2 percent target has triggered declines in the yen.
"This comment raises questions about (the BOJ's) implicit preference for a weaker yen," said Richard Franulovich, senior currency strategist at Westpac Banking Corp in New York.
Data showing a 15 percent contraction in Chinese exports in March, the worst in about a year, also supported the dollar against the euro as well as the Australian dollar.
The Aussie dollar, considered a liquid proxy of China plays because of the two countries' trade links, hovered near a low of $0.7553 hit earlier in the session, bringing the Aussie back within sight of a six-year low of $0.7534 set earlier in the month.
The euro was last down 0.1 percent against the dollar at $1.0577 after earlier hitting a nearly 1-month low of $1.05205. The dollar was last down 0.1 percent against the yen at 120.04 yen.
The dollar was down nearly 0.5 percent against the Swiss franc at 0.9769 franc after earlier hitting its highest in over three weeks against the franc of 0.98625 franc.
The dollar index, which measures the greenback against a basket of six major currencies, was last up 0.11 percent at 99.45.