Russian President Vladimir Putin, the focus of intensifying diplomatic activity this week, didn't immediately respond to the new conditions. Sustained violence in eastern Ukraine is driving a frenzy of high-level meetings in Brussels, Paris and Normandy aimed at reconciling Russia and Ukraine.
Putin is holding his first face-to-face meetings with Western leaders: Cameron and French President Francois Hollande on Thursday night, and German Chancellor Angela Merkel on Wednesday. There's hope that Putin could also talk with Ukraine's new president-elect.
Obama and Cameron spelled out new demands for Russia at a joint news conference after a Group of Seven world leader summit. Putin was meant to have hosted the summit in Sochi, but the G-7 countries cancelled that after his aggressive moves in Ukraine, and met without him in Brussels.
It was the group's first summit in two decades without the participation of Russia.
The US and Europe, after imposing economic sanctions on Russia in recent months, are considering toughening them.
To avoid even harsher sanctions, Cameron said Putin must meet three conditions: recognize Petro Poroshenko's election as the new leader in Kiev, stop arms from crossing the border and cease support for pro-Russian separatist groups concentrated in eastern Ukraine.
"If these things don't happen, then sectoral sanctions will follow," Cameron said. "The next month will be vital in judging if President Putin has taken these steps. And that is what I will urge President Putin to do when I meet him later today."
Obama said the G-7 leaders unanimously agree with the steps Cameron outlined. But they weren't so explicit in written statements issued after two days of meetings, and an Obama aide later described the potential sanctions in different terms than Cameron.
"If Mr Putin takes those steps, then it is possible for us to begin to rebuild trust between Russia and its neighbors and Europe," Obama said. "We will have a chance to see what Mr. Putin does over the next two, three, four weeks, and if he remains on the current course, then we've already indicated that kinds of actions that we're prepared to take."
Obama acknowledged that so-called sectoral sanctions, which would hit key sectors of Russia's economy, could have a bigger impact across Europe because of European economic ties to Russia, and said he didn't necessarily expect all European countries to agree on them. But, he said, "it's important to take individual countries' sensitivities in mind and make sure that everybody is ponying up."
"My hope is, is that we don't have to exercise them because Mr Putin's made some better decisions," Obama said.
It's unclear how Putin will react to the conditions, and whether he could claim that he has already fulfilled them.
Putin has already welcomed the Ukrainian election and said Russia would "respect the will of the Ukrainian people." On sealing the border and withdrawing support to separatists, no one has caught Putin red-handed, and he insisted in a French television interview Thursday that the US has "no proof" that Russians are playing any role at all in the unrest in eastern Ukraine.
Obama said he thought the fact Putin didn't immediately denounce the outcome of Ukraine's election last month offers hope he's moving in a different direction. "But I think we have to see what he does and not what he says," he added.
There's one key leader Putin is not scheduled to meet this week — Obama. Amid the worst East-West crisis in a generation, US-Russia relations are so tense that the French president is eating back-to-back dinners Thursday night so that Obama and Putin don't have to share the same table.
Obama said they may have an opportunity to talk when they both attend events commemorating the 70th anniversary of the D-Day invasion Friday in Normandy.
Putin's meetings with Cameron, Hollande and another with Merkel on Friday illustrate how Western countries are divided over how to deal with the Russian leader.
France's president said that Putin could meet with Poroshenko when both come to Normandy along with Obama and other leaders, to mark 70 years since the D-Day Allied invasion that helped wrest Western Europe from Hitler's grip.
European leaders are hoping the famous beaches, which saw the turning point in World War II, may now serve as a diplomatic platform to end the Ukraine-Russia crisis.
"It is an exceptional international meeting that must serve the cause of peace," Hollande said.
The top US and Russian diplomats met in Paris on Thursday, and agreed that Ukraine should be a bridge between Russia and the West and not be a "pawn" in a power struggle.
"We would like to see Ukraine peaceful, stable. A place for all those who live in Ukraine... to be feeling equal, respected, and listened to, living in peace being a bridge, not being a pawn," Russian Foreign Minister Sergey Lavrov said after meeting US secretary of state John Kerry.
Despite fundamental differences over the March annexation of Crimea and unrest in eastern Ukraine since, Russia has signaled its readiness for a direct dialogue with Poroshenko, a billionaire candy tycoon, who was elected May 25.
The election, which came three months after pro-Moscow President Viktor Yanukovych was chased from office by crowds following months of street protests and allegations of corruption, was seen as a critical step toward resolving Ukraine's protracted crisis.
Since his ouster, Russia has annexed the Crimean Peninsula in southern Ukraine, the eastern regions of Luhansk and Donetsk have declared their independence from Kiev, and the interim Ukrainian government has launched an offensive in the east to quash an uprising that has left dozens dead.