(Updates throughout, changes byline, dateline)
* Euro, Italian bonds rebound on budget deficit cut report
* European shares rise, led by Italian banks; U.S, to open firmer
* Indian, Indonesian, Turkish currencies hit by oil and inflation pressure
* Oil holds near four-year highs
By Sujata Rao
LONDON, Oct 3 (Reuters) - The euro bounced off six-week lows on Wednesday, European shares rose and Italian bonds rallied as some of the worries that have rippled across markets this week were soothed by signs Rome was amenable to cutting budget deficits and debt in coming years.
A report in the Corriere della Serra newspaper -- later confirmed to Reuters by a government source -- said the deficit would fall to 2.2 percent of gross domestic product in 2020 and to 2 percent in 2021 from the 2.4 percent earlier outlined.
That brought relief to markets which had fretted that Italy's decision to expand budget deficits well beyond what was agreed by a previous government would deepen its debt problems while stoking conflict with the European Union, whose officials have already expressed concerns.
The euro, which had hit a six-week trough of $1.1506 after suffering five straight days of losses, firmed 0.3 percent while Italian borrowing costs eased off 4-1/2-year highs, after jumping 50 basis points since budget details emerged last Thursday.
Two-year yields fell 15 bps.
"It's all about Italy today. The news about a possible reduction in budget deficits has been positively taken by all markets, and the euro is back up too -- that's the main risk parameter for euro crisis concerns," said Bernd Berg, global macro and FX strategist at Woodman Asset Management in Zurich.
A pan-European equity index opened a quarter percent higher , while the Milan bourse jumped more than one percent . The moves were led by an initial 3.1 percent bounce in Italian banks. The banks' huge government debt holdings make them particularly vulnerable to bond selloffs and has pressured Italian stock markets for months.
The moves pushed down the premium investors demand for holding Italian risk relative to that of safer Germany to around 290 bps, down from a five-year high over 300 bps on Tuesday and sapped some demand for safe-haven assets such as German bonds, the dollar and Swiss franc.
The euro jumped half a percent to the franc and 0.4 percent to the yen . Analysts noted however that other southern European markets, Spain and Portugal, which were caught up in Greece's 2010-2011 crisis, had been resilient to spillover from Italy this year.
"Contagion has been far more subdued than in the past and that's an important signal. I expect policymakers to latch on to that as proof (Italy's crisis) is self-imposed and not a reflection of what's happening as a whole in the weaker (euro zone) countries," said Salman Ahmed, chief investment strategist at Lombard Odier in London.
He added that Italy's high yields -- two-year bonds there pay 1.3 percent versus zero in Portugal -- were also enticing some investors.
Wall Street too appeared set for a firmer open, with futures for the S&P 500 the Dow up around 0.2 percent.
Broader world market sentiment remains jittery though, partly because of the bellicose rhetoric still emerging from the coalition government in Rome, but also due to fears that a Sino-U.S. tariff row would escalate once China reopens after a week-long holiday.
While U.S. President Donald Trump agreed a new trade pact with Mexico and Canada, a controversial clause in the trilateral agreement, forbidding similar deals with "non-market" countries, was seen as raising risks for Sino-U.S. talks.
Volatility in U.S. tech shares is also making investors wary, with Facebook down almost 6 percent in the past three sessions after disclosing its worst security breach ever.
World shares were flat near two-week lows while MSCI's index of Asia-Pacific shares outside Japan slipped 0.2 percent and Japan's Nikkei closed 0.7 percent lower.
China's financial markets are closed and will resume trade on Oct. 8. That's the day Trump and Chinese President Xi Jinping are due to attend G-20 meetings, meaning news could emerge on the trade front.
The dollar index, measuring the greenback against a basket of major currencies, pulled back from six-week highs of 95.744.
Emerging currencies derived some relief from the weaker dollar but the Indian rupee hit a new record low and the Indonesian rupiah touched a more than 20-year low, as oil prices near four-year highs weighed on importing nations' finances.
Indonesian authorities stepped in to support the currency and India is expected to raise interest rates this Friday.
Turkey's lira also weakened one percent to the dollar as the country, another oil importer, posted data showing inflation at more than 25 percent. That increases pressure on the central bank to raise rates again after a whopping 625 bps move in August.
Woodman Asset Management's Berg said however the worst looked to be over for Turkey, adding: "the currency has appreciated and the central bank has hiked rates aggressively so...towards the end of the year, inflation numbers should stabilise and to me, the worst of the currency crisis is over."
Brent crude inched up to $84.89 per barrel, close to four-year highs of $85.45 touched earlier this week.
Additional reporting by Swati Pandey in Sydney