* U.S.-Mexico agreed deal on migration on Friday
* Yen, euro sag as dollar recovers across the board
* Chinese offshore yuan a whisker from 2019 low
* Graphic: World FX rates in 2019 http://tmsnrt.rs/2egbfVh
By Tommy Wilkes
LONDON, June 10 (Reuters) - The Mexican peso surged more than 2% on Monday 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 boosted the dollar and knocked the safe-haven yen lower.
Foreign exchange investors had rushed for the safety of the Japanese yen in recent weeks after U.S. President Donald Trump's threat to slap tariffs on Mexico shook investor confidence.
On Monday, the dollar staged a recovery as investors dumped the yen and euro. Some analysts said the U.S. currency should rebound as investors realised their assessment of the path for Federal Reserve interest rate cuts was overly dovish.
The dollar's index rallied 0.3% to 96.824. Against the euro it rose 0.3% to $1.1304, with the single currency also under pressure after sources told Reuters that European Central Bank policymakers were open to cutting rates if economic growth weakened.
The Mexican peso rose more than 2% to 19.2275 pesos per dollar after trading resumed for the first time following Friday's migration agreement. (nL2N23E0MX) (nL2N23G03P)
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.
"While we should expect 'risk-on' type trading in the early part of the week, we are very cautious over the sustainability of any rebound in risk appetite," MUFG analysts said in a note, citing caution about Mexico's backing for the deal's details and worries Trump may push other trade disputes.
The yen shed 0.4% to 108.65 after earlier hitting its weakest since late May, though it remains 3.3% stronger than its levels of April.
Bart Wakabayashi, Tokyo branch manager at State Street Bank, said the lift to sentiment from the U.S.-Mexico deal would "probably spill over to optimism with China and hopefully some progress there."
"We've had trade talks with the EU, with Japan. Hopefully these will start to turn to the positive narrative," he said.
The yuan brushed its lowest since late November after weak import data that reignited worries about slowing Chinese domestic demand. The offshore rate was slightly lower at 6.9488 yuan per dollar but held above Friday's low.
The euro pulled back from 2-1/2 month highs hit on Friday in the wake of an ECB meeting that investors viewed as less dovish than expected.
The ECB did not - as some had anticipated - hint at rate cuts in the face of a deteriorating economic outlook, instead saying that rates would stay "at their present levels" until mid-2020.
But on Sunday two sources told Reuters that policymakers were open to cutting the ECB's policy rate again if economic growth weakened and a strong euro hurt the bloc.
Sterling was caught by the dollar's recovery, losing 0.2% to $1.2709. (Additional reporting by Daniel Leussink in Tokyo; editing by John Stonestreet)