* Euro zone bond yields fall further
* Australia cuts key interest rate
* U.S. rate cut speculation grows
* EZ flash inflation weaker-than-expected
* Italian bond yields tumble Euro zone periphery govt bond yields tmsnrt.rs/2ii2Bqr (Updates pricing)
By Dhara Ranasinghe
LONDON, June 4 (Reuters) - A stellar rally in euro zone bond markets extended to riskier Italian and longer-dated maturities on Tuesday on growing speculation of rate cuts from major central banks and data showing inflation in the bloc has slowed more than expected.
With benchmark 10-year German bond yields stuck near record lows, investors snapped up other countries' bonds and riskier longer-dated ones.
France's 30-year bond yield slid 5 bps to its lowest since 2016 at 1.18%, Germany's 15-year bond yield fell to just 0.007% - a whisker from negative territory.
Italian bonds extended their rally, with analysts saying that speculation about a generous package of cheap multi-year bank loans from the ECB - which could be unveiled at Thursday's ECB meeting - also boosted peripheral debt markets.
"Underlying inflation continues to disappoint and we think the ECB will downgrade its forecasts in terms of inflation on Thursday," said Chris Scicluna, head of economic research at Daiwa Capital Markets.
"What matters for monetary policy is inflation so the onus is on the ECB to provide stimulus, perhaps in the form of a generous TLTRO," he said, referring to the ECB's cheap loan scheme.
Data showed euro zone inflation fell to 1.2% in May from 1.7% in April. In a potentially more worrying sign, underlying inflation or prices excluding volatile food and energy prices - the ECB's preferred measure - fell to 1.0% from 1.4% a month earlier.
Germany's 10-year Bund yield fell to minus 0.219% , matching Monday's record low.
Dutch and French 10-year bond yields hit fresh multi-year lows .
Analysts said signals from global central banks only fuelled speculation about the ECB's next steps, adding to the downward momentum in world bond yields.
Australia's central bank on Tuesday cut its cash rate to a record low 1.25% in what could be the first in a series of stimulus measures.
St. Louis Federal Reserve president James Bullard said on Monday that a U.S. interest rate cut "may be warranted soon" given risks to global growth posed by trade tensions and weak U.S. inflation.
Investors have ratcheted up rate-cut bets sharply in the past two weeks. Market pricing suggests more than two Fed rate cuts are priced in for this year.
And money market futures now price in roughly a 50% chance of a 10 basis-point ECB rate cut by the end of the year .
"I think we will see a pretty dovish message from (ECB chief Mario) Draghi this week," said Mark Dowding, a senior portfolio manager at BlueBay Asset Management.
"The appearance of downside risks means the ECB will want to convey to investors that the scope for monetary policy action is not exhausted and they stand ready to support the economy and inflation if conditions weaken."
In Italy, comments from Prime Minister Giuseppe Conte after Monday's market close boosted bond markets and banking stocks .
Conte said the government had to abide by EU budget rules until such time as they could be changed. He also told his two coalition partners to end their constant feuding or seek new elections, threatening to resign.
Italy's 10-year bond yield fell to a two-month low at 2.48%.
Reporting by Dhara Ranasinghe, editing by John Stonestreet and Ed Osmond