Tensions started to spiral last year when Trump pulled out of a deal the U.S. and other world powers signed with Iran during the Obama administration.
The Russian President said that he had urged Iran not to quit the 2015 agreement.
If it chooses to restart the programme, it could resume mass enrichment at its main facility in the central Iranian town of Natanz.
Trump told the Jewish community that the US now has "left the horrible Iran nuclear" deal.
President Donald Trump has dramatically increased pressure on Tehran, withdrawing from an international agreement aimed at ending its nuclear programme.
Pompeo said that countries like India, if it gets the exemption, would be asked to bring down their oil imports from Iran to zero in six months' time.
The sanctions that'll come into force Monday penalize countries that don't stop importing Iranian oil and foreign companies that do business with blacklisted Iranian entities.
The United States has moved to reimpose sanctions that had been lifted under the landmark deal to curb Iran's nuclear program and has vowed to punish foreign firms that do business with Iran.
While Trump has not totally ruled out meeting with Rouhani on the sidelines of the General Assembly, the Iranian president appears in no mood to meet his US counterpart.
Their delicate, and perhaps unrealistic, task is to build a case for Tehran to respect the deal's limits on its nuclear program even though Washington has pulled out.
Trump in May withdrew from the accord between Iran and six world powers aimed at stalling Tehran's nuclear capabilities in return for lifting some sanctions.
President Hassan Rouhani has faced mounting pressure from lawmakers over his handling of an economic crisis, partly triggered by the withdrawal of the United States from the 2015 nuclear deal.
Iran has asked the International Court of Justice to order the United States to lift the sanctions, reimposed after US President Donald Trump pulled out of a multilateral 2015 accord.
Khamenei said talks should continue with Europe, which is trying to salvage the 2015 agreement despite the withdrawal of the United States.
Iran has argued that US President Donald Trump breached a 1955 treaty with his decision to reimpose the sanctions after withdrawing from a multilateral nuclear accord.