"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.
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.