President Obama and Iran's new president have a historic opportunity at the United Nations meeting Tuesday to mark a breakthrough in the two countries' long, difficult relationship.

After decades of hostility between the U.S. and Iran since the 1979 hostage crisis, every interaction between the two leaders carries broader meaning. Will the two shake hands or sit down together for a one-on-one meeting?

The White House has been peppered with questions after newly elected Iranian President Hassan Rouhani suggested a reset of relations, calling for “constructive” dialogue in a Washington Post op-ed last week.

Administration officials though have stressed a cautious approach ahead of the United Nations gathering.

“I don't think anything will happen by happenstance on a relationship this important,” said White House Deputy National Security Adviser Ben Rhodes, when asked Monday if the two presidents would shake hands.

The White House is still not saying whether the two leaders will meet. Although there is no meeting on Obama's schedule right now, the White House left the door open to the possibility.

Press secretary Jay Carney said last week that Obama was "willing" to have a meeting but said that Tehran must show it was serious about resuming nuclear talks.

Iran has made a concerted effort to tone down its rhetoric since Rouhani took office in August, and Tehran’s new leader has promised not to use Iran's nuclear energy program to develop weapons.

The White House may be waiting to make a decision about a meeting after watching Rouhani's speech before the U.N. general assembly Tuesday. Rouhani’s predecessor Mahmoud Ahmadinejad regularly delivered fiery addresses condemning the United States, leading American and Israeli officials to walk out.

Rouhani campaigned on a message of “hope and pragmatism,” promising to help ease U.S.-led sanctions to boost Iran's economy.

Ahead of the address, both Democratic and GOP lawmakers are warning Obama to react cautiously to Tehran’s outreach.

Even if Rouhani is sincere, he is not the country's leader, and Congressional lawmakers remain skeptical of Iran’s supreme leader Ayatollah Ali Khamenei.

Sens. Lindsey Graham, R-S.C., and Robert Menendez, D-N.J., the chairman of the Foreign Relations Committee, penned a letter to Obama Monday urging him to maintain a tough stance against Iran’s nuclear program.

“Like you, we viewed the election of Hassan Rouhani as an indicator of discontent amongst the Iranian people, and we have taken note of recent diplomatic overtures by Iran,” they wrote. “However, whatever nice words we may hear from Mr. Rouhani, it is Iranian action that matters. We would welcome a credible and verifiable agreement with Iran.”

Some of Obama's critics say his decision to halt a planned strike on Syria after leader Bashar Assad crossed a “red line” by using chemical weapons will only embolden Iran, which is a much bigger threat to U.S. interests.

Israeli Prime Minister Benjamin Netanyahu plans to warn the United Nations that Iran's diplomatic offensive could be a trap.

Obama, who also is scheduled to speak Tuesday at the U.N., will try to strike a delicate balance on Iran, and the related issues of Syria's chemical weapons and ongoing peace talks between Israel and the Palestinians.

Although the U.S. is open to diplomatic solutions to issues in the Middle East, top administration officials said they will rely on former President Reagan's mantra: Trust but verify.

Obama’s U.N. address also gives him another opportunity to rally international support against Assad if he fails to follow through on a U.S.-Russian brokered plan to disarm his chemical arsenal. The U.S. is pushing the United Nations Security Council to pass a binding resolution calling for the destruction of Assad's chemical stockpile.

Iran's leaders are pursuing a swift agreement to end sanctions on their economy, seizing on a recent letter from Obama to the new leader last month that promised relief if Tehran demonstrated a willingness to “cooperate with the international community, keep your commitments and remove ambiguities,” according to a report in the New York Times.

The Obama administration has welcomed Rouhani's more positive message.

“But the fact of the matter is, actions are what are going to be determinative here,” White House spokesman Josh Earnest said Friday. “The Iranians, for a number of years now, have been unwilling to live up to their obligations to the international community as it relates to their nuclear program."

After the United Nations meeting is over Tuesday afternoon, Obama will return to promoting his health care reform overhaul. Obama will attend former President Bill Clinton's Global Initiative event in New York where the two presidents will “engage in a conversation about the benefits and positive impacts of health care reform in America and access to quality health care around the globe,” the White House said.

The conversation is part of the administration's efforts to ramp up promotion of the health care law before exchanges open for enrollment Oct. 1. A White House official said the conversation would kick off an “aggressive six-month public awareness effort.”