(CNN) -- U.S. National Security Adviser Susan Rice warned on Sunday that it "would be a grave mistake" if Russian President Vladimir Putin intervened militarily in the ongoing crisis in Ukraine.
Rice, who spoke on NBC's "Meet the Press," was among U.S. leaders saying they want to see a unity coalition government in the country after President Viktor Yanukovych fled Kiev, the capital, and a unanimous vote in Parliament removed him from power.
"The United States is on the side of the Ukrainian people," Rice said. The people expressed themselves peacefully, she said, and Yanukovych "turned on" the people by using violence against them.
A former Soviet republic closely allied with Russia, Ukraine is home to many people who speak Russian and identify with Russia.
David Remnick, editor of The New Yorker, recently returned from Russia and reported from the country for many years. On "Fareed Zakaria GPS" on Sunday, Remnick said that Putin considers Ukraine and Russia "linked by blood."
"If you look through Putin's eyes, specifically, this is his area of interest. It's not the United States and even not Europe," Remnick said. "This is really complicated for us."
Rice said the U.S. wants to see constitutional reform, democratic elections and a unity government, and President Barack Obama stressed that to Putin in a recent conversation.
"Putin was in agreement" at that time, Rice said.
Republican Sen. John McCain appeared on CBS's "Face the Nation" with strong words for Putin.
A partition of Ukraine would "not be acceptable," McCain said.
"The Ukrainian people will determine their own future. They want to be Western. ... They do not want to be Eastern," he said.
If he were Putin, he said, he would be nervous about the protest movement happening in Ukraine.
Appearing on "Fox News Sunday," Sen. Dick Durbin, D-Illinois, said he spoke with former Prime Minister Yulia Tymoshenko on Saturday just after she was released from prison following a unanimous vote in Parliament.
Durbin said she assured him that she's "looking for a peaceful resolution to the problems in Ukraine, to follow the constitution and the law."
Sen. Kelly Ayotte, R-New Hampshire, said there must be focus on forming a unity government. "Yanukovych needs to step aside, and I will say this: Now that the Olympics are over, we need to watch the behavior of the Russians," she said.
Obama "needs to up his game and send a clear, unequivocal, public message to Putin not to interfere in what is happening in Ukraine," Ayotte said, "to let the Ukrainian people determine their future, to ensure that there is no interference in their sovereignty."
Durbin agreed that more pressure should be placed on Putin.
"We have to combine our efforts with the European Union to help Ukraine move forward in a peaceful democratic way. We have to put the pressure on Putin to stop his efforts to undermine this natural evolution towards democracy," Durbin said, adding that he thinks Yanukovych "has always been a puppet of Moscow."