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* Futures off: Dow 0.03%, S&P 0.06%, Nasdaq 0.16%

Jan 2 (Reuters) - U.S. stock index futures slipped on Tuesday, setting a somber tone for the first trading day of 2024, as Apple dipped on a report of a broker downgrade and investors pondered if last year's big market gains could be sustained.

The three major U.S. stock indexes notched monthly, quarterly and annual gains on Friday as traders priced in higher chances of interest rate cuts from the Federal Reserve this year on the back of cooling inflation.

For 2023, the benchmark S&P 500, the tech-heavy Nasdaq and the blue-chip Dow posted double-digit gains. The S&P 500 closed on Friday within 1% of its record closing high reached on Jan. 3, 2022.

However, U.S. stock futures came under pressure on Tuesday, with Apple falling 1.2% in premarket trading after a report from thefly.com said Barclays downgraded the iPhone maker's stock.

Meanwhile, the yield on 10-year U.S. Treasury notes , a benchmark for global borrowing costs, ticked above 4.0000% to a two-week high earlier in the session before easing to 3.9425%.

At 5:42 a.m. ET, Dow e-minis were down 10 points, or 0.03%, S&P 500 e-minis were down 3 points, or 0.06%, and Nasdaq 100 e-minis were down 27.5 points, or 0.16%.

S&P Global's final reading of U.S. manufacturing activity for December is due at 9:45 a.m. ET. Other economic data this week includes weekly jobless claims, monthly private and non-farm payrolls data as well as services sector data.

Stocks of companies linked to cryptocurrencies, such as Coinbase Global and MicroStrategy, jumped 6.0% and 9.2%, respectively, as bitcoin stormed above $45,000 for the first time since April 2022 on optimism around the possible approval of exchange-traded spot bitcoin funds.

Energy majors such as Exxon Mobil and Chevron gained more than 1% each, tracking a jump in crude prices after a naval clash in the Red Sea raised the chances of Middle East supply disruptions. (Reporting by Sruthi Shankar in Bengaluru; Editing by Devika Syamnath)