The single currency rocketed last week after the ECB did not - as some had anticipated - hint at interest rate cuts, instead saying rates would stay "at their present levels" until mid-2020.
But on Sunday two sources familiar with the ECB's policy discussions said a cut was firmly in play if the bloc's economy was to stagnate again after expanding by 0.4pc in the first quarter of the year.
"The policymakers are trying to figure out if there is anything they can do to ease (further) and whether it would work," said Kit Juckes, an analyst at Societe Generale.
However, Juckes said it was hard to see the euro rallying significantly from here unless investors slashed their short positions or European economic fundamentals improved.
The euro fell 0.2pc to $1.1306 after hitting $1.1348 last week, its highest since March.
The dollar index gained 0.3pc to 96.825.
The greenback had weakened last week after poor economic data encouraged investors to scale up their bets that the Federal Reserve would soon cut interest rates.
The Mexican peso surged more than 2pc after the United States and Mexico struck a deal on migration to avert a trade tariff war, supporting a rebound in investor risk appetite that also knocked the safe-haven yen lower.
Goldman Sachs analysts noted that trade tensions had generally pushed the dollar higher this year and last.
"But even if trade disputes worsen from here, we are unsure that this would imply sustained dollar appreciation. Instead, the various pillars of dollar support seem to be giving way, and we now see the greenback on a choppy downward path through year-end," they said in a note to clients.
The Mexican peso rose more than 2pc to 19.186 pesos per dollar after trading resumed for the first time following Friday's migration agreement.
US President Donald Trump had threatened to impose 5pc import tariffs on all Mexican goods starting on Monday if Mexico did not commit to do more to tighten its borders.
The yen shed 0.3pc to 108.65 after earlier hitting its weakest since late May, though it remains more than 3pc stronger than its levels of April. Investors also sold the Swiss franc .
China's yuan brushed its lowest since late November after weak import data reignited worries about slowing domestic demand.
The offshore rate was slightly lower at 6.9532 yuan per dollar but held above Friday's 2019, low.
Sterling was caught by the dollar's recovery and a surprise 0.4pc contraction in British gross domestic product in April.
The pound fell 0.4pc to $1.2689.