However, those efforts hit a roadblock on Tuesday, when Iran rejected an offer by President Obama for a one-on-one meeting with Rouhani at the United Nations.
That public change of heart by Rouhani on Tuesday raised serious questions on Iran's commitment to reestablish ties with the U.S., according to the Republican Senators.
Those questions, they added, extend far beyond the diplomatic impasse between Washington and Tehran on Iran's nuclear enrichment program.
"While Iran’s pursuit of a nuclear weapons capability is our top concern, it is not our only concern," they said.
Iran's continued sponsorship of Hezbollah, as well as its military support to embattled Syrian President Bashar Assad.
"We must see transparent, tangible, and verifiable steps by the Iranian regime to fulfill its international obligations" on all fronts, before the White House can begin rebuilding its ties with Iran, they said.
"The American people and Congress will not support anything less." the Senators added.
In August, the Treasury Department's Terrorism and Financial Intelligence directorate levied a series of economic sanctions against top commanders from Hezbollah's "Unit 1800."
The unit is tasked with providing weapons, equipment and training to radical militant groups affiliated with the Lebanese terrorist organization around the world, with the explicit backing of the Iranian government, according to a department statement issued Thursday.
Most recently, Unit 1800 members have been actively supporting militants looking to destabilize the democratic government in Iraq and bolster the Assad regime.