A meeting hasn't been planned but could happen, the senior officials told The Wall Street Journal.
Rouhani and President Obama are both scheduled to address the General Assembly in New York Tuesday.
No U.S. president has met a top Iranian leader since Islamic radicals overthrew the pro-U.S. Shah Mohammad Reza Pahlavi in 1979.
"I think it's fair to say that the president believes there is an opportunity for diplomacy when it comes to the issues that have presented challenges to the United States and our allies with regards to Iran, and we hope that the Iranian government takes advantage of this opportunity," White House spokesman Jay Carney told reporters Wednesday after saying there were "currently no plans" for the two presidents to meet at the General Assembly.
Carney said Obama indicated to Rouhani in a recent letter exchange the United States was "ready to resolve the nuclear issue in a way that allows Iran to demonstrate that its nuclear program is for exclusively peaceful purposes."
"The letter also conveyed the need to act with a sense of urgency to address this issue," Carney said, "because as we have long said, the window of opportunity for resolving this diplomatically is open, but it will not remain open indefinitely."
Rouhani told NBC News in an interview at Iran's presidential compound Wednesday he considered the tone of Obama's letter "positive and constructive" and said the letter exchange "could be subtle and tiny steps for a very important future."
Rouhani also told the network his country would not, under any conditions, build nuclear weapons.
"We have, time and again, said that under no circumstances would we seek any weapons of mass destruction, including nuclear weapons, nor will we ever," he said through an NBC translator.
Iran is "solely seeking peaceful nuclear technology," Rouhani said.
It was his first interview with a U.S. news organization since winning election in June against five hard-liners on a promise to end Iran's international isolation.
On taking office Aug. 3, Rouhani said, "The Iranian people voted 'yes' to moderation."
In the six weeks since then, Rouhani, a Shiite Muslim cleric, and his Cabinet have sent signals Western diplomats see as indicating a willingness to improve relations with the West and ease economic woes besetting Iranians, due to years of U.S. sanctions because of the disputed nuclear program.
"We have heard a lot in the world from President Rouhani's administration about its desire to improve the government of the Islamic Republic of Iran's relations with the international community -- and President Obama believes we should test that assertion," Carney said.
Obama told Spanish-language Noticias Telemundo Tuesday despite the need to test Iran's sincerity, he saw "an opportunity ... for diplomacy."
"I hope the Iranians take advantage of it," Obama said.
The United States alleges Tehran is well along toward developing an atomic weapons program. Iran insists its program is for peaceful purposes and is therefore legally entitled to enrich uranium.
Iran's supreme leader, Ayatollah Ali Khamenei, who was Iran's president in the 1980s, has said nuclear weapons development would conflict with Islamic values.
Rouhani's address the U.N. General Assembly is widely expected to be conciliatory and strike a softer note than that of his predecessor, Mahmoud Ahmadinejad, British newspaper The Daily Telegraph reported.
Ahmadinejad regularly used the forum as a pulpit for inflammatory speeches, railing against Israel and the United States.
Western diplomats regularly walked out during his addresses.
Iranians have expressed hope Rouhani or another senior Iranian figure will meet with U.S. counterparts on the sidelines of the annual world leaders gathering at the General Assembly.
The Journal said the Obama administration was preparing for high-level meetings next week with the Iranian government.