LONDON, Jan 4 (Reuters) - Euro zone bond yields ticked higher on Thursday after data showed inflation climbed in France and in Germany's states and the downturn in the private sector was less deep than initially thought in December.

Figures on Thursday showed French inflation rose slightly in December as economists expected, with the year-on-year rate at 4.1%, compared to 3.9% in November.

Inflation across Germany also climbed, according to regional data which comes before the countrywide figures at 1300 GMT (2 p.m. CET).

Yields, which move inversely to prices, had been trading around 3 or 4 basis points (bps) lower across the board but were last up for the day.

The German 10-year yield, the benchmark for the euro zone, was up 3 basis points (bps) at 2.042%. It hit a 1-year low of 1.896% last week.

France's 10-year bond yield was last 2 bps higher at 2.578%, up from an 11-month low of 2.395% touched last week.

HCOB's Composite Purchasing Managers' Index, a survey-based gauge of the euro zone's economic health, was revised up for December to match the November reading but remained below the line separating growth from contraction.

Adrian Prettejohn, Europe economist at Capital Economics, said the figures did not meaningfully change the narrative around the economy and inflation.

"We expect the euro-zone economy to stagnate in the first half of the year, with only a slow recovery after that. And we think inflation will trend towards 2% by the middle of the year," he said.

Global bond yields tumbled in November and December as inflation in the U.S. and Europe slowed more than expected and central banks signalled their rate-hiking cycles were almost certainly over. That caused investors to bet on big interest rate cuts next year.

Yet yields have risen in the new year as markets have reined in those rate cut bets slightly and taken a more cautious tone as they confront a busy economic calendar.

Germany's two-year bond yield, which is sensitive to European Central Bank interest rate expectations, was last up 2 bps at 2.429%.

U.S. figures on Wednesday showed that job openings and the number of people quitting their posts fell to the lowest in around three years, suggesting the pressure on the labour market is falling.

The minutes from the Federal Reserve's December meeting, also released on Wednesday, showed officials launched a debate on rate cuts, with some voicing fears about how long the economy can withstand the current high borrowing costs.

Traders on Thursday expected around 160 bps of rate cuts from the ECB this year, up slightly from just under 170 at the same time last week, according to money market pricing. (Reporting by Harry Robertson; Editing by Hugh Lawson)