* Energy stocks drop as oil prices slide

* Weekly jobless claims stuck at higher levels

* Apple, Microsoft, Amazon lower

NEW YORK, Sept 10 (Reuters) - U.S. stocks closed lower after a choppy trading session on Thursday as heavyweight tech-related stocks resumed their decline following a sharp rebound the previous session, while elevated jobless claims reminded investors of a still-difficult recovery ahead.

Names that have rallied since March lows, such as Apple Inc , Microsoft Corp and Amazon.com, all fell.

Tesla Inc rose slightly, initially helping to limit the Nasdaq's losses before the tech-heavy index's slide widened.

The NYSE FANG+TM Index, which includes the core FAANG stocks, also fell, and all 11 sectors of the S&P 500 traded lower.

Wall Street's main indexes bounced back sharply on Wednesday from their biggest three-day rout since March, as investors returned to tech-focused stocks that are deemed insulated from the current economic downturn.

The number of Americans filing new claims for unemployment benefits remained high last week, Labor Department data showed, as layoffs and furloughs persisted across industries.

In addition, the U.S. Senate on Thursday killed a Republican bill that would have provided around $300 billion in new coronavirus aid, as Democrats seeking far more funding prevented it from advancing.

Senate Majority Leader Mitch McConnell said in an interview with Fox News the electoral outcome for control of the Senate could go either way.

"It’s more of this sort of stew of stuff than has come together than any one particular thing you can point to, and maybe most importantly you are looking at an extended market," said Stephen Massocca, senior vice president at Wedbush Securities in San Francisco.

"This thing has had a massive move so the line of least resistance might be to correct a little bit."

Massocca cited election uncertainty and the length of time it may take the economy to fully reopen due to the coronavirus pandemic.

Unofficially the Dow Jones Industrial Average fell 406.29 points, or 1.45%, to 27,534.18, the S&P 500 lost 60.07 points, or 1.77%, to 3,338.89 and the Nasdaq Composite dropped 221.97 points, or 1.99%, to 10,919.59.

The S&P tech index was one of the weaker performers on Thursday afternoon. Despite the recent pullback, the tech index is up about 24% in 2020, far outperforming the benchmark S&P 500's rise of 3.5% in the same period.

Many market participants view the sell-off as a bout of turbulence rather than the start of a deeper slide.

The CBOE volatility index climbed on Thursday. The index hit a near three-month high at the start of a historically tumultuous September. Investors have also remained cautious as data paints a mixed picture of U.S. economic health.

A separate report showed U.S. producer prices rose slightly more than expected in August as the cost of services increased solidly.

Energy stocks dropped as oil prices extended losses after U.S. data showed a surprise build in crude stockpiles last week and on forecasts for lower global oil demand.

(Reporting by Chuck Mikolajczak and Herb Lash in New York Editing by Matthew Lewis)