The main U.S. indexes closed up more than 1% on Monday as oil prices fell after President Donald Trump said he had ordered the military to postpone strikes against Iranian power plants following "productive conversations" with Tehran.

However, Iran's Parliamentary Speaker Mohammad Baqer Qalibaf posted on social media that no talks had been held with the U.S., contradicting Trump's announcement that there were talks between the United States and Iran in the past day in which the two sides had "major points of agreement" and that a deal could be done soon to settle the war.

While U.S. equities fell last week, they staged a sharp recovery on Monday after Trump's comments sent oil prices lower. Equities had been trading lower earlier in the day after threats of attacks on Israeli and Iranian power networks.

"You never know who to believe but it does appear that Trump is trying to start discussions with somebody in Iran to resolve the war despite strong denials from Iran. This has caused significant optimism in stock prices today with the market up strongly although off its highest levels because of the Iranian denials," said Tim Ghriskey, senior portfolio strategist at Ingalls & Snyder in New York.

According to preliminary data, the S&P 500 gained 73.77 points, or 1.13%, to end at 6,580.25 points, while the Nasdaq Composite gained 300.94 points, or 1.39%, to 21,948.55. The Dow Jones Industrial Average rose 631.06 points, or 1.39%, to 46,211.53.

The CBOE Volatility Index, Wall Street's fear gauge, retreated after earlier hitting its highest level in two weeks.

Oil prices settled down more than 10% on Monday. All of the S&P 500's 11 major industry sectors advanced with big gains in cyclical sectors such as consumer discretionary while advances in defensive sectors such as healthcare and consumer staples were weaker.

"The volatility is likely to continue and it's all about the price of oil. Nothing else really matters to people in the short term. So when oil prices are down, stocks go up and vice versa," said Bob Doll, chief investment officer at Crossmark Global Investments. "What's up the most today is not a surprise. It's things with economic sensitivity."

Meanwhile, investors trimmed bets for an interest-rate hike from the U.S. Federal Reserve to a roughly 12% probability for December from 25% in the prior session, according to CME Group's FedWatch.

Traders were betting on a 70.8% chance that rates would be unchanged by year end after scaling back bets for cuts last week after the central bank struck a hawkish tone due to concerns about higher inflation.

The small-cap Russell 2000 was outperforming large-cap indexes on Monday. On Friday, the index, which is sensitive to higher interest rates, had ended more than 10% below its record close of January 22, confirming it had been in correction territory.

Fuel-hungry airlines jumped on Monday as the price of oil fell with Alaska Air, American Airlines and United Airlines all gaining. Shares in cruise ship operators soared with Norwegian Cruise Line, Carnival Corp and Viking Holdings all gaining.

Banks, which had weakened during the conflict, gained ground on Monday with the S&P 500 Banking index showing its biggest daily gain since before the war.

Investors will look forward to Fed speakers, business activity surveys and consumer sentiment readings this week.

In individual stocks, Synopsys rallied after activist investor Elliott Investment Management has a multibillion-dollar investment in the electronic design automation firm.