* European stocks fall on anti-euro rhetoric
* Italian bond yields rise, bank stocks sink
* Oil near 4-year highs
* U.S. stocks set to open lower
* Graphic: World FX rates in 2018 tmsnrt.rs/2egbfVh
By Helen Reid
LONDON, Oct 2 (Reuters) - Stocks fell worldwide and European assets sold off on Tuesday after anti-euro comments from an Italian lawmaker dented the single currency and sent Italy's bond yields to multi-year highs, despite attempts by the government to backtrack.
A boost to investors' risk appetite from the new U.S.-Mexico-Canada trade pact proved short-lived with the MSCI world equity index falling back 0.3 percent.
U.S. futures traded down 0.3 percent to 0.4 percent ahead of Wall Street's open, while PepsiCo gained in pre-market trading after reporting stronger-than-expected profit.
The leading index of euro zone stocks lost 0.7 percent and the pan-European STOXX 600 fell 0.5 percent as Italian assets came under renewed pressure, though the sell-off lost steam by midday.
The economics spokesman for Italy's ruling League party, Claudio Borghi, said in a radio interview that most of the country's problems could be solved by having its own currency.
His comments drove Italian 10-year bond yields to a new 4-1/2-year high, pushing the spread between Italian and German yields to the widest for more than five years.
Shares in Italian banks, which have large sovereign bond holdings, hit a 19-month low before recovering part of their losses.
But Borghi and Prime Minister Giuseppe Conte backed down, calling the euro "unrenounceable", helping to calm markets and erasing losses for Italy's FTSE MIB.
"While our economists do not expect systemic implications for the global economy, contagion risks have risen," Goldman Sachs analysts said.
"We think European risky assets remain vulnerable and there is potential for negative spillovers to the Euro area given the high trade exposure to Italy."
The euro fell 0.6 percent to its lowest since Aug. 21 at $1.1504, then recovered to trade down 0.3 percent at $1.1542.
The single currency has been hurt by concerns that a significant increase in the Italian budget will deepen Italy's debt and deficit problems, and by extension the European Union's.
"The history of the euro zone tends to be one of great fudges - think of the case of Greece," the Saracen global income and growth fund's manager, David Keir, said.
"But I would caution against any wider systemic spreading. The reality is making kneejerk reactions to big political decisions can very much be the wrong thing to do," he said.
Asian stocks were lower as the lift from an agreement that saved the North American free-trade deal faded.
China's financial markets are closed for the week of Oct. 1-5 for national holidays, but data showing weaker factory growth in China also hit Hong Kong stocks.
The U.S. and Canada forged a last-minute deal on Sunday to salvage NAFTA as a trilateral pact with Mexico, rescuing a $1.2 trillion open-trade zone that had been about to collapse after nearly a quarter century in operation.
The trade pact helped the dollar index rise to its highest since Aug. 21, at 95.744. It was last up 0.3 percent at 95.546.
The dollar's strength weighed on the leading emerging markets stock index, which fell 1.3 percent, setting it on course for its biggest one-day loss for a month.
Oil prices recoiled slightly as investors took profit after crude surged to near four-year highs in the previous session.
Crude contracts rose to their highest since November 2014 as the deal to salvage NAFTA stoked economic growth expectations, with impending U.S. sanctions on Iran seen raising prices.
On Tuesday U.S. crude futures, having touched a high of $75.91, were trading flat at $75.33 a barrel.
Brent crude edged down 0.4 percent to $84.60 a barrel, having rallied the previous day to $85.45.
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Reporting by Helen Reid Editing by Louise Ireland