LONDON, June 5 (Reuters) - Germany's 10-year government bond yield fell for a third straight day on Wednesday after data suggested the U.S. labour market is slowing, with focus set to shift to the European Central Bank's policy announcement on Thursday.

A report on private payrolls showed U.S. hiring slowed to a four-month low in May, adding to recent signs that the economy is cooling and pulling global bond yields lower. The first rate cut from the Bank of Canada in four years was also helping the mood in bond markets.

However, a closely watched purchasing managers' index survey later showed that the U.S. services sector snapped back strongly to growth last month, causing yields to rise again somewhat.

The German 10-year bond yield, the benchmark for the euro zone bloc, was last down 3 basis points (bps) at 2.513%, around its lowest in two weeks. Yields move inversely to prices.

Yields have fallen in recent days as data from the United States, including a weak consumer spending report last week, has suggested the economy might finally be slowing enough to allow the Federal Reserve to cut interest rates.

"U.S. yields have been falling for four days straight on growth pessimism," said Benjamin Schroeder, senior rates strategist at ING. "Bund yields have been dragged lower too."

European yields tend to move on U.S. data thanks to the size and importance of the American economy and other central banks' wariness of straying too far from the Fed.

Germany's two-year bond yield, which is more sensitive to interest rate expectations, was 1 bp lower at 2.984%.

The ECB is widely expected to lower its deposit rate from a record high of 4%, but there remains uncertainty about the future path for rates.

Money market traders are pricing around 64 basis points (bps) of cuts this year, implying two quarter-point moves and around a 50% chance of a third cut.

"We're all sitting and waiting for tomorrow. We think the ECB will do the 25 (basis point cut) and then say we have to wait and see how data evolves," said Jens Peter Sørensen, director, fixed income research at Danske Bank.

"If they do that then I think the market reaction should be fairly benign."

Some policymakers have tried to take a move at the following meeting in July off the table, while others, including French rate-setter Francois Villeroy de Galhau, appeared more open to a second straight move.

Italy's 10-year yield was 4 bps lower at 3.825%, meaning the yield gap between Italian and German bonds , a measure of risk premium investors seek to hold Italian paper, stood at 130 bps. (Reporting by Samuel Indyk; Editing by Andrew Heavens, Sriraj Kalluvila and Emelia Sithole-Matarise)