LONDON, June 5 (Reuters) - Euro zone government bond yields rose on Monday after a sharp fall last week, while the gap between Italian and German borrowing costs narrowed to its smallest since April 2022.

Germany's 10-year yield, the euro zone's benchmark, was last up 7 basis points (bps) at 2.38%. It fell 23 bps last week as data showed that euro zone inflation cooled more than expected in May.

Analysts said a few factors were likely pushing up euro zone bond yields on Monday, including Friday's strong U.S. jobs report which kept the pressure on the Federal Reserve to raise rates at least one more time from 5% to 5.25% level.

"Employment in the U.S. remains strong and the Fed's pause is likely to be challenged for as long as it will be the case," said Florian Ielpo, head of macro at Lombard Odier AM.

Ielpo also said Monday's rise in oil prices - driven by Saudi Arabia's new plan to further cut production - was a reminder that "the inflation fight is not over just yet". Brent crude was last up around $1.30 a barrel at $77.45.

Meanwhile, European Central Bank President Christine Lagarde acknowledged "signs of moderation" in core inflation on Monday but she reaffirmed it was too early to call a peak in that key gauge of price growth.

Italy's 10-year yield was 9 bps higher at 4.07% on Monday.

The gap between Italy and Germany's 10-year yields - which is closely watched as a sign of investor sentiment towards the euro zone's more indebted countries - briefly fell to 160 bps in early trading, the lowest since April 2022. It was last at 167 bps.

Analysts have said the narrowing in the spread has been driven by both economic and technical factors.

"Euro area non-core bond spreads are expected to widen slightly amid a challenging growth environment and an accelerated (ECB) Quantitative Tightening," said Florian Spate, bond strategist at Generali Investments in an e-mailed comment.

"The development is forecast to remain orderly," he added.

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The slowdown in euro zone inflation has caused traders to marginally reduce their bets on further European Central Bank rate hikes, which are typically more painful for countries with higher debt loads.

ECB euro short-term rate (ESTR) forwards still price in two 25 bps rate hikes by summer, with the September 2023 forward at 3.68%, from 3.75% late last week.

Italy's economy has been performing better than expected, with the yields on offer on the country's bonds remaining some of the highest in Europe and therefore attractive to investors.

Olivier De Larouziere, chief investment officer for global fixed income at BNP Paribas Asset Management, said last month's decisions by ratings agencies to hold Italy's credit rating steady have reassured investors.

"The bias is clearly towards an upgrade after a very long period of downgrades," he said, noting that Italy was managing its debt more effectively than investors had expected.

Germany's 2-year bond yield, which is sensitive to interest rate expectations, was last up 8 bps at 2.92%, after falling 13 bps the previous week.

Survey data on Monday showed that euro zone business activity expanded at the slowest pace in three months in May, as the service sector cooled and the manufacturing downturn continued. Markets showed little reaction to the data. France's 10-year bond yield was up 6 bps at 2.92% after rating agency S&P left the country's AA rating untouched on Friday.

(Reporting by Harry Robertson, additional reporting by Stefano Rebaudo; Editing by Emelia Sithole-Matarise, Ed Osmond and Alex Richardson)