* Sources say ECB open to rate cuts, euro drops
* U.S.-Mexico migration deal boosts sentiment
* Chinese offshore yuan a whisker from 2019 low
* Graphic: World FX rates in 2019 tmsnrt.rs/2egbfVh (Adds details, analyst quotes)
By Tommy Wilkes
LONDON, June 10 (Reuters) - The euro pulled back from 2-1/2 month highs on Monday as a U.S.-Mexico deal over migration boosted the dollar and after sources said European Central Bank policymakers were open to cutting the ECB's policy rate should economic growth worsen.
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 0.4% 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 0.2% to $1.1306 after hitting $1.1348 last week, its highest since March.
The dollar index gained 0.3% 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 2% 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 2% to 19.186 pesos per dollar after trading resumed for the first time following Friday's migration agreement.
U.S. President Donald Trump had threatened to impose 5% import tariffs on all Mexican goods starting on Monday if Mexico did not commit to do more to tighten its borders.
The yen shed 0.3% to 108.65 after earlier hitting its weakest since late May, though it remains more than 3% 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 0.4% contraction in British gross domestic product in April. The pound fell 0.4% to $1.2689. (Editing by John Stonestreet, editing by Ed Osmond)