* Czech's December PMI falls

* PMIs for Hungary, Brazil, Mexico on watch

* Turkey's lira inch closer to 30 per dollar mark

* Stocks down 0.6%, FX off 0.4%

Jan 2 (Reuters) - Emerging markets currencies and stocks opened 2024 on a weak footing as investors were cautious ahead of key U.S. economic data, while factory activity readings across Turkey, Poland and Czech Republic fell.

The MSCI's gauge of emerging market stocks was down 0.6% by 0930 GMT, while a basket of currencies dipped 0.4% against the dollar.

The currencies index is set for its worst day since early September if losses hold.

Emerging markets assets had a robust 2023 as risk sentiment notched higher after the U.S. Federal Reserve's dovish stance prompting bets on reduced borrowing costs.

Investors await U.S. job openings and non-farm payrolls due later this week for further clues on the outlook for interest rates.

Across Asia, heavyweight China stocks led broader equity declines. Shanghai's Composite index fell 0.4%, while the blue-chip stocks and Hong Kong shares slid over 1%.

A private sector survey showed China's factory activity expanded at a quicker pace in December, but business confidence for 2024 remained subdued.

The survey contrasted with official purchasing managers' index (PMI) data released on Sunday which showed manufacturing activity shrinking at a faster pace and more than expected in December.

"It is worth noting that the Caixin and NBS (National Bureau of Statistics of China) PMIs moved in opposite directions in both November and December, sending somewhat confusing signals about the economy’s near-term growth momentum," J.P.Morgan economists said in a note.

"Demand-supply imbalance remains the key concern," they added.

Turkey's lira touched fresh lows of 29.7105 against the dollar, inching closer to the 30 per dollar mark after declining nearly 37% last year.

Data showed December PMIs across Turkey, Poland and Czech Republic fell in December.

The Polish zloty edged up 0.4%, while the Czech's crown ticked up 0.2%.

PMIs for Hungary, Brazil and Mexico are due later on Tuesday.

The Russian rouble fell as much as 90.70 against the dollar, hitting a one-month low.

The Bank of Israel lowered short-term borrowing rates for the first time in nearly four years on Monday.

Israel's Supreme Court on Monday struck down a highly disputed law passed by Prime Minister Benjamin Netanyahu's right-wing government which had rolled back some of the high court's power and sparked months of nationwide protests.

Stock markets across Romania and Russia were closed on account of New Year holidays.

For GRAPHIC on emerging market FX performance in 2023, see http://tmsnrt.rs/2egbfVh

For GRAPHIC on MSCI emerging index performance in 2023, see https://tmsnrt.rs/2OusNdX

For TOP NEWS across emerging markets

For CENTRAL EUROPE market report, see

For TURKISH market report, see

For RUSSIAN market report, see (Reporting by Siddarth S in Bengaluru; editing by Jason Neely)