LONDON, Jan 2 (Reuters) - European stock markets fell on Tuesday, changing course after having initially opened higher as traders bet on central banks cutting rates in 2024, while oil prices rose after a naval clash in the Red Sea.

Global stock markets rose overall in 2023, having gained particularly in the last two months of the year, while bond yields fell. This upbeat market sentiment initially continued on Tuesday as traders returned from end-of-year holidays, with the pan-European STOXX 600 hitting its highest in nearly two years in early European trading. Euro zone bank stocks rose to their highest since 2018.

But markets struggled to hold on to the gains, and by 1210 GMT the STOXX 600 was down 0.2% on the day. London's FTSE 100 was down 0.2% and Germany's DAX was down 0.2%.

The MSCI World Equity index was down 0.3% on the day.

Wall Street futures pointed to declines in U.S. stock indexes, with Nasdaq futures down 1% and S&P 500 futures down 0.7%.

Traders are watching for clues as to whether major central banks will think inflation has come down enough to allow for rate cuts.

"There is a feeling that (monetary) easing is coming and it seems like there is more to go in the rally in the short term," said Nordea chief analyst Jan von Gerich. "I think there's a risk to the downside for stocks but the momentum is strong right now," he said.

Oil prices jumped more than 2%, in a move analysts said was due to an escalation in tensions in the Red Sea as well as hopes for strong demand from China, where investors are expecting fresh stimulus measures.

U.S. helicopters repelled an attack on Sunday by Iran-backed Houthi militants on a Maersk container vessel in the Red Sea, sinking three Houthi boats and killing 10 militants. Investors are weighing up the risks of the Israel-Gaza war becoming a wider regional conflict, which could close crucial waterways for oil transport.

Brent crude rose 2.4% to $78.85 a barrel while U.S. West Texas Intermediate crude was at $73.41 a barrel, up 2.4%.

Separately, the head of energy firm E.ON said instability in the Middle East could send energy prices soaring, but that Germany's gas supply situation is far better than it was after Russia cut off supply last winter.

Data pointing to subdued business confidence in China for 2024 weighed on Chinese assets during Asian trading. China's onshore blue chip index was down 1.3% and Hong Kong's Hang Seng index fell 1.5%.

The U.S. dollar index was up around 0.6% at 101.98, strengthening to a 12-day high after it lost roughly 2% last year on bets that U.S. rates will come down.

The U.S. 10-year Treasury yield, which gained overall in 2023, rose to 3.9576%.

Economic data due later in the week includes the U.S. non-farm payrolls report on Friday which could provide clues as to the U.S. Federal Reserve's next move. Minutes from the last Fed meeting in December are also expected to give insight into central bankers' thinking regarding rate cuts.

At its December policy meeting, the Fed adopted an unexpectedly dovish tone and forecast 75 basis points in rate reductions for 2024. Other major central banks, including the European Central Bank (ECB) and Bank of England (BoE), have indicated that they will hold rates higher for longer.

In Europe, flash inflation figures for the euro zone are due on Friday, which RBC Capital Markets analysts said in a note are "likely to be the most significant additional data point prior to the January ECB meeting".

"Anything barring a large increase in inflation would represent a significant surprise," the analysts said.

The euro fell around 0.7% against the dollar to$1.0966.

Euro zone government bond yields rose, with the benchmark 10-year German yield up 6 basis points on the day at 2.091%.

Gold was up 0.2% at $2,066.1 an ounce.

(Reporting by Elizabeth Howcroft and Dhara Ranasinghe; editing by Jason Neely and David Evans)