Iran "will not withdraw an iota from the definite rights of people," Rouhani was quoted by the state news agency IRNA as saying on Tuesday.
Enriching uranium is the core issue at the center of the nuclear controversy and a potential pathway to atomic weapons. Iran says it needs some enrichment for a Tehran research reactor.
Rouhani also suggested that his trip to the U.N. along with Foreign Minister Mohammad Javad Zarif in coming weeks, his first visit to the West since his inauguration in August, could be a new chance for progress. The United Nations has scheduled Rouhani's speech for Sept. 24, hours after President Barack Obama is expected to address the annual General Assembly.
"Maybe in coming weeks, the first talks on the nuclear case will take place in New York," Rouhani told a group of clerics.
Talks between Iran and world powers were last held in April amid deadlock over Western efforts to rein in Tehran's nuclear program.
Rouhani in his remarks quoted by IRNA urged the West to change its policies, including withdrawing economic sanctions imposed on Iran over the nuclear program. He said "the job" of reaching a nuclear accord "will not be done without respect."
"There will be no achievement through imposing sanctions and pressure on the Iranian nation," Rouhani said.