He made the remarks during a press conference on Italian capital, Rome, president.ir reported.
There is no obstacle in Iran in the way of investment of American investors, he said, but the key lies in Washington, the Americans should solve the problem if there is any.
It is up to US to improve relations with Iran, Rouhani said.
“Non-hostile and tension-free relations between Iran and the US are possible; the key to this lies in Washington not in Tehran; and they should use the key, put aside hostilities and pressure on Iranian people,” Rouhani said, adding that US officials “should understand that the era of sanctions and pressure has ended.”
On his first official visit to Europe, Rouhani also took a swipe at Saudi Arabia, saying its military campaign in neighboring Yemen was a failure and a frustration.
Rouhani is midway through a four-day trip to Italy and France following the signing last year of a nuclear accord with world powers and the lifting of financial sanctions.
While EU firms are lining up to sign lucrative business deals, the United States is keeping some of its sanctions in place.
Rouhani also said he would be happy to see US businessmen in Iran.
"I would like to see the Americans set aside their hostility and chose another way, but inside the US there are some problems, there is no unified voice," he said, noting that "the Zionist lobby" was "very influential".
He also rejected accusations that Iran was funding terror organizations. "It is clear that Iran is a country opposed to terrorism and a country that fights terrorism," he said.
"The Americans know very well that when it comes to important regional issues they cannot achieve anything without Iran's influence or say," Rouhani said, speaking through a translator.
Adding to tensions in the region is the recent deterioration in relations between Iran and Saudi Arabia. Riyadh broke off diplomatic ties with Tehran this month in an escalating row over the Saudi beheading of the Shia cleric Sheikh Nimr al-Nimr.
Rouhani said Saudi Arabia’s beheading of Sheikh Nimr earlier this month was a big mistake.
Answering a question from reporters, President Rouhani said Saudi officials should apologize for the killing of Nimr, the massacre of the Yemeni people, the killing of a large number of people during last year’s Hajj rituals, and the support of terrorists active in the region.
Regarding the assaults on Saudi diplomatic missions in the Iranian capital, Tehran, and the city of Mashhad following the killing of Nimr, the Iranian president said Iran has fulfilled all of its duties to reduce tensions.
Saudi Arabia announced on January 2 that it had executed Sheikh Nimr, an outspoken critic of the regime in Riyadh. The execution caused international outrage and sparked anti-Saudi demonstrations in many countries, including Iran.
Demonstrations were held in front of the Saudi Embassy in Tehran and its consulate in Mashhad. Some angry protesters mounted the walls of the consulate in Mashhad.
“We condemned the raid on the embassy, (and) identified and arrested the assailants… [They] will be tried. But Saudi measures in continuing tensions are unjustifiable,” President Rouhani said.
The Iranian president also said that the US administration can take the opportunity to enhance economic relations with Iran following the implementation of Tehran’s landmark nuclear agreement with six world powers.