The market received Trump's comments with cautious relief after the turbulence of recent weeks, although Iran later denied that the parties had held any discussions.
The Dow Jones industrial average rose 1.4 percent, the broad S&P 500 index rose 1.2 percent and the tech-heavy Nasdaq composite index closed up 1.4 percent.
The price of oil has dropped significantly and a barrel of North Sea oil (Brent) is now trading between $95 and $100.
Shares of companies with high fuel costs rose on Monday. For example, cruise company Norwegian Cruise rose 6.2 percent, while airlines American Airlines and United Airlines rose 3.6 and 4.5 percent, respectively.
Corrected: In a previous version, the wrong figure was stated for how much American Airlines had risen.
