Rouhani made the comments in his first interview with a U.S. news outlet since his election in June. Ann Curry, an NBC News correspondent, conducted the interview at Rouhani's presidential compound in Iran’s capital.
Rouhani “struck a conciliatory tone,” NBC News said.
The interview will air on the network’s nightly newscast Wednesday. A few excerpts were released ahead of the broadcast.
President Obama recently exchanged a series of letters with the Iranian president, and could lay a foundation for the two to rekindle their ties.
"It could be subtle and tiny steps for a very important future," Rouhani told Curry. "I believe the leaders in all countries could think in their national interest and they should not be under the influence of pressure groups. I hope to witness such an atmosphere in the future."
Though the two countries terminated their diplomatic relationship in 1980, the two leaders seem to be open to talks. Iran and the U.S. were close allies for nearly four decades , until the 1979 Iran hostage crisis.
After Rouhani’s election in June, The White House said it would be open to engaging “directly” with Iran on the issue of its alleged nuclear weapons program. Rouhani's predecessor, former President Mahmoud Ahmadinejad, was viewed by the U.S. as an enemy.
Both the new Iranian leader and Obama are scheduled to appear at the United Nations’ General Assembly next week.
Obama administration officials, however, have said there are “currently no plans” for the two leaders to meet at the U.N., but that could change.
Obama will discuss Iran’s alleged nuclear program with Israeli Prime Minister Benjamin Netanyahu at the White House on Sept. 30, in addition to the situation in Syria. Israel hinted last year it could launch a preemptive attack on Iran to defend itself against Tehran’s nuclear weapons.
Ahmadinejad has said that Israel must be “wiped off the map.”
Tehran has never admitted to producing nuclear weapons.
The International Atomic Energy Agency, which is part of the U.N., released a quarterly report in late August that revealed Iran continues to expand its nuclear program.
According to the report, Iran recently installed 1,000 advanced uranium enrichment centrifuges in one city, the report said. But the agency also concluded that the stock of uranium that can be further enriched to weapons-grade material only grew by a small amount.
Iran has also been a hot topic lately because of the recent developments in Syria. Secretary of State John Kerry said while meeting with Netanyahu last Sunday in Jerusalem that if Syria is not held accountable for its chemical weapons, it would embolden Iran.
The U.N. Security Council, meanwhile, will now hash out a diplomatic deal between the U.S. and Russia to eliminate Syria’s chemical weapons stockpile.
This story was updated at 4:57 p.m.