(Updates with U.S. markets' closing levels)
* MSCI global stock index eases
* Copper rises amid wage talks at world's largest mine
* Oil prices end higher
By Caroline Valetkevitch
NEW YORK, June 5 (Reuters) - The Nasdaq hit a record closing high for the second day in a row on Tuesday, helped by tech and consumer discretionary shares, while the euro recovered after Italy's new prime minister said the government had never considered leaving the euro zone.
Shares of Amazon, Apple and Netflix all rose. Upbeat economic data also helped the S&P 500 eke out a small gain.
But gains in equities were limited by worries about trade discussions. White House economic adviser Larry Kudlow said President Donald Trump may seek separate talks with Canada and Mexico in a bid to get individual trade deals with the two countries.
Mexico responded to Trump's metals tariffs by imposing its own duties on American steel, while also targeting agricultural products from pork to bourbon.
Trump's administration imposed tariffs of 25 percent on imported steel and 10 percent on aluminum in March, citing national security grounds.
The market "continues to focus on one of two big issues - where we're headed with the trade negotiations - NAFTA, China and tariffs with Europe. That's probably the biggest concern for investors. But, on the positive side, (are) the economy's strength and earnings' strength," said Rick Meckler, partner at Cherry Lane Investments, a family investment office in New Vernon, New Jersey.
The U.S. services sector activity accelerated in May, pointing to robust economic growth in the second quarter.
The Dow Jones Industrial Average fell 13.71 points, or 0.06 percent, to 24,799.98, the S&P 500 gained 1.93 points, or 0.07 percent, to 2,748.8 and the Nasdaq Composite added 31.40 points, or 0.41 percent, to 7,637.86.
MSCI's gauge of stocks across the globe shed 0.13 percent, while the pan-European FTSEurofirst 300 index lost 0.32 percent.
Comments by Italian Prime Minister Giuseppe Conte caused the dollar to fall and reverse gains to a near six-month high, as the euro recovered.
The euro rose 0.1 percent against the dollar to $1.1712 . The dollar index, tracking it against a basket of major currencies, fell 0.15 percent.
The Mexican peso gained 0.10 percent versus the U.S. dollar at 20.43. The Canadian dollar rose 0.01 percent versus the greenback at 1.30 per dollar.
Earlier, the Mexican peso and Canadian dollar fell against the dollar as the latest developments in the growing trade conflict between the United States and its neighbors prompted selling in the currencies.
U.S. Treasury yields fell as traders piled back into lower-risk government debt, however, after Conte vowed to enact economic policies that could add to the nation's debt load.
Benchmark 10-year U.S. Treasury notes last rose 3/32 in price to yield 2.9259 percent, from 2.937 percent late on Monday.
Copper hit a six-week high as concerns about the potential supply impact of wage negotiations at the world's biggest copper mine helped push prices back above $7,000 a tonne.
Copper rose 2.04 percent to $7,117.00 a tonne.
Oil prices ended higher. U.S. crude rose 77 cents to settle at $65.52 a barrel, while Brent gained 9 cents to settle at $75.38.
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Reporting by Caroline Valetkevitch in New York Additional reporting by Richard Leong, Sinead Carew and Gertrude Chavez-Dreyfuss in New York, Medha Singh in Bengaluru, Marc Jones and Helen Reid in London and Danilo Masoni in Milan; Editing by David Gregorio and James Dalgleish