US President Donald Trump said the United States and Iran were holding productive talks and that Iran wanted to reach an agreement.
Trump said the discussions had continued throughout the day.
Iranian officials, however, were described as repeatedly denying that negotiations with Washington were taking place.
Trump said an agreement could be implemented as soon as the following day or the day after, while adding that he had substantial time available.
He warned that Iran would face severe consequences if no agreement was reached.
Unnamed regional and US officials said a proposal under discussion would provide safe passage for 60 days through the strait between Iran and Oman.
Those officials said no toll would be charged under the proposed arrangement and that its duration could be extended.