The euro held around $1.13 and analysts said a stronger-than-forecast German business confidence survey on Monday was buoying the single currency.
But German 10-year bund yields remained below zero and that worried global stock markets despite some steadying on Tuesday.
The euro has remained in a fairly narrow range of $1.12-$1.16 in 2019 despite a marked slowdown in the euro zone economy that has prompted fresh stimulus from the European Central Bank.
"We expect EUR/USD to stabilize around the current level of 1.13 and see a limited downside for the rest of week," said currency strategists at ING.
The dollar rebounded modestly after global markets recoiled on Monday in the wake of an inversion in the U.S. Treasury yield curve, which has signalled a recession in the past.
Meanwhile the Swiss franc, which tends to strengthen during times of economic and political turmoil, neared a 20-month high versus the euro of 1.12.
Appreciation in the so-called safe-haven franc is being driven by risk aversion as a result of concerns about the global economy and could prompt the Swiss National Bank (SNB) to intervene to weaken the currency.
"In case of a no-deal Brexit the 1.10 level is likely to be reached very quickly again. And in fact we would then expect the SNB to intervene in the markets, preventing a stronger collapse of the EUR-CHF exchange rate," said Ulrich Leuchtmann, head of forex research at Commerzbank in Frankfurt
"Even franc appreciation to 1.10 against the euro is likely to be annoying enough considering the SNB's efforts to stabilise inflation in positive territory."
The franc on Monday was marginally weaker versus the euro at 1.1234.
Sterling was effectively flat at $1.3197 GBP=D3 after spending the previous day confined to a narrow range when British lawmakers wrested control of the parliamentary agenda from the government for a day in a highly unusual bid to find a way through the Brexit impasse.
They will now vote on a range of Brexit options on Wednesday, giving parliament a chance to indicate whether it can agree on a deal with closer ties to the European Union.