LONDON, July 2 (Reuters) - Euro zone government bond

yields were slightly lower on Tuesday after data showed euro

zone inflation cooled in June, as the market steadied after

sharp rises the previous day.

Germany's 10-year bond yield, the benchmark for

the euro zone bloc, was down one basis point (bp) at 2.59%.

Figures showed that euro zone inflation fell to 2.5% in June

from 2.6% in May, as economists expected.

Yet core inflation - which strips out volatile food and

energy costs - came in at 2.9%. That was unchanged from May and

above expectations for a fall to 2.8%.

"It already seemed unlikely that the European Central Bank

would cut interest rates at its meeting in July, and June's

inflation data will reinforce policymakers' inclination to move

very cautiously," said Jack Allen-Reynolds, deputy chief euro

zone economist at Capital Economics.

Benchmark 10-year U.S. Treasury yields dipped on Tuesday

following jobs market data and comments by Federal Reserve Chair

Jerome Powell that monetary policy is closer to the tradeoff

point where rate cuts are appropriate.

The euro zone is "very advanced" on the disinflationary path

but there remain "question marks" over the economic outlook,

European Central Bank President Christine Lagarde said.

Italy's 10-year yield was down 3.5 bps at 4.07%,

around its highest in three weeks. The gap between Italian and

German bond yields dropped to 146 bps.

Germany's two-year bond yield, which is more

sensitive to ECB rate expectations, was down 1.5 bps at 2.91%.

Bond yields rose sharply on Monday as a rally in safe-haven

bonds, driven by French President Emmanuel Macron's gamble in

calling snap elections in early June, unwound after the results

of the first round came in. Bonds are seen as more stable than

stocks and preferable at times of stress.

Marine Le Pen's far right Rassemblement National party

comfortably won the first round. Analysts said the market was

comforted by signs that the most likely outcome is a hung

parliament, however, prompting a rally in stocks and the euro.

The second round of voting is on Sunday and the final shape

of the French parliament will depend on horse-trading between

the groups opposing Le Pen.

France's 10-year yield was 2 bps lower at 3.33%.

The gap between France and Germany's 10-year yields - a

gauge of the risk premium investors demand to hold French debt -

ticked down to 72 bps.

It fell on Monday as the results came in, after rising to

its highest since 2012 at 85 bps last week.

Policymaker Pierre Wunsch said it is a relatively easy

decision to cut rates again, but after that it depends on more

progress on inflation.