By Raf Sanchez, Washington
11:42PM BST 03 Aug 2013
"Until we see a significant slowdown of Iran’s nuclear activities, we believe our nation must toughen sanctions and reinforce the credibility of our option to use military force at the same time as we fully explore a diplomatic solution to our dispute with Iran," the senators write.
The Obama administration has taken a wait-and-see approach to Mr Rouhani, who won a surprise election victory after promising to end years of economic turmoil caused by Western sanctions.
Congress, however, is moving ahead with an aggressive new round of sanctions.
The House of Representatives voted this week 400-20 for measures designed to put pressure on the few remaining buyers of Iranian oil and to choke off Tehran's access to its dwindling foreign exchange reserves.
The Senate is expected to pass its own bill in September ahead of the next expected round of diplomatic negotiations between Iran and the six-nation bloc of the US, Britain, France, Germany, Russia and China.
In the letter to President Barack Obama, the senators note that Mr Rouhani has "pledged re-engagement" with the international community but warn "Iran has used negotiations in the past to stall for time".
"We urge you to bring a renewed sense of urgency to the process," the senators write to Mr Obama. "We need to understand quickly whether Tehran is at last ready to negotiate seriously."
A senior Senate aide said that the letter, which was spearheaded by Senators Robert Menendez and Lindsey Graham, was intended to temper the optimism of some US officials that Mr Rouhani's inauguration would lead to a diplomatic breakthrough.
"Soft noises of Tehran changing its tune is one thing; a concrete action of stepping back from the nuclear precipice is a much different thing," he said.
Both Iranian and Western negotiators are focused on two competing timelines.
The first is how long until Iran reaches a "critical nuclear capability", the capacity to produce weapons grade uranium so quickly it would be undetectable to western intelligence.
The second is how long Iran's economy can withstand the severe sanctions regime and especially the stranglehold it places on access to foreign currency reserves needed to prop up the Iranian rial.
"The central question is which comes first: Iran reaching economic collapse or Iran reaching critical nuclear capability?" said Mark Dubowitz, executive director of the Foundation for Defense of Democracies think tank.
"The single most important piece of intelligence for the US may not be about Iranian nuclear physics - it may have more to do with Iranian economics."
Analysts and Western intelligence are constantly working to produce estimates of both timelines.
A major report by the Institute for Science and International Security this month suggested that Iran's nuclear programme is likely to reach a critical stage by June 2014.
Its economic situation is difficult to assess but under the current sanctions regime, Iran is widely believed to have at least a few years of foreign currency reserves left.
US lawmakers hope that by stepping up current sanctions they can increase economic pressure and reduce the amount of time Iran has left to negotiate.
"There is a broad consensus that to get a negotiated deal with the Iranians, massively intensified sanctions are needed to accelerate the date at which their economy goes over the cliff," said Mr Dubowitz.
The small group of representatives who opposed the House sanctions bill this week argued that the US should use Mr Rouhani's inauguration as a chance to reset the relationship with Iran.
"Why aren’t we at least curious to find out whether or not President Rouhani means that he wants to pursue this course of peace?" said Representative Keith Ellison, a Democrat.
Some analysts have suggested, however, that the threat of new sanctions could strengthen Mr Rouhani's hand if he intends to try to convince Ayatollah Ali Khamenei, Iran's supreme leader, to negotiate.