Also on the agenda are nuclear-armed North Korea, which set off another round of short-range missiles in tests earlier Monday, NATO expansion, the situation in Georgia after its conflict with Russia last year, human rights and arms control.
A senior State Department official traveling with Clinton said she intends to speak to Lavrov and Medvedev about "what specific forms of pressure Russia would be prepared to join us and our allies in if Iran fails to live up to its obligations."
The official said it was critical to get tangible signs of support from Moscow because the more united the international community is the more likely pressure on Iran is to work. The official spoke on condition of anonymity to discuss the pending private diplomatic discussions.
On Sunday, Clinton warned Iran that the world "will not wait indefinitely" for evidence it is not trying to develop atomic weapons.
She said a recent meeting in Geneva in which Iran and six world powers resumed nuclear talks was "a constructive beginning, but it must be followed by action" from the Iranians.
Iran insists it has the right to a full domestic nuclear enrichment program that it maintains is only for peaceful purposes, such as energy production.
It is already under three sets of U.N. sanctions. Russia and China have balked at imposing new penalties on Iran although Medvedev has hinted the Russian position might be shifting after Tehran disclosed a previously secret uranium enrichment site near the holy city of Qom.
But U.S. officials believe it will be a hard sell to convince the Russians on fresh penalties since Iran agreed to allow U.N. inspectors to visit the Qom site and has agreed, in principle, to send most of its low-enriched uranium to Russia for reprocessing.
Iran agreed to allow inspections of the Qom site following the Geneva talks between Iran's chief nuclear negotiator and diplomats from the U.S., Russia, China, Britain, France and Germany.
The Iranians were given time to decide whether to accept a package of incentives in exchange for Iran's compliance with international demands to suspend its uranium enrichment or face new sanctions.
The Obama administration is anxious not to let up on the pressure and Clinton will be looking for Russian expressions of support for sanctions and other penalties should Iran continue to refuse by the end of the year, the officials said.
As Clinton left Washington on Friday for Switzerland, Britain, Ireland and Russia, Medvedev said his government does not want to see any more nations develop nuclear weapons, signaling that Moscow shares U.S. concerns about Iran. But he said nothing about potential sanctions.
Clinton's visit to Moscow is her first since becoming America's top diplomat and since President Barack Obama, who visited Russia in July, vowed to "reset" U.S.-Russia relations.
The senior official traveling with Clinton said that there had been some improvements in cooperation, including a recent agreement that allows U.S. military planes to transport lethal materiel over Russia to Afghanistan.
But he acknowledged numerous disagreements remained.
"We still have differences on Georgia, we still have differences on NATO, we still have differences even on some of the areas that we are working together on, including Iran, and that's cause for further discussion," the official said.
Negotiators from the two countries are racing to reach agreement on a successor to the 1991 Strategic Arms Reduction Treaty, or START I, and Clinton wants to underscore the urgency of the talks, they said.
She also will explore possible cooperation on missile defense following Obama's decision not to proceed with Bush-administration plans to base such a system in eastern Europe. Russia had vehemently opposed those plans and has welcomed Obama's new approach.
Though the Kremlin is pleased with the move, it has pushed for a link between missile defense and the START talks. Russian and U.S. diplomats were to hold consultations about missile defense on Monday before Clinton's arrival.
Clinton will also join Lavrov in chairing a meeting of a commission set up by Obama and Medvedev to improve cooperation and coordination on a variety of matters, including Afghanistan. Russian Prime Minister Vladimir Putin is traveling to Beijing on a trade mission and will not be in Moscow for the talks with Clinton.