Sen. Chris Murphy, D-Conn., a member of the same committee, said, "I think the time is now to rapidly ratchet up our sanctions, whether it's on Russian petrochemical companies or on Russian banks."
"If Russia does get away with this, I do think that there's a potential that a NATO ally is next. And, yes, there will be economic pain to Europe (under tightened sanctions). But it's time for them to lead as well."
President Barack Obama has said his administration is prepared to take further action against Russia if diplomatic efforts to destabilize the conflict fail.
Vice President Joe Biden plans to be in Kiev, the Ukrainian capital, on Monday and Tuesday to meet with government leaders and democracy advocates.
Defense Secretary Chuck Hagel said last week that the U.S. is looking for ways to reassure its NATO allies of its strong commitment to collective defense. The Pentagon's press secretary, Rear Adm. John Kirby, has said that American officials are considering a range of additional measures to bolster air, maritime and ground readiness in Europe.
Corker, who plans to be in the region in May, said unless the Russians "immediately begin moving the 40,000 troops on the border which are intimidating people in Ukraine, unless they begin immediately moving them away, I really do believe we should be sanctioning some of the companies in the energy sector, Gazprom and others. I think we should hit some of the large banks there."
Geoffrey Pyatt, the U.S. ambassador to Ukraine, said that an international agreement forged late last week designed to ease tensions in Ukraine may be "the best chance that we have got to achieve a diplomatic de-escalation of this crisis. And we're working hard at it."
He told CNN's "State of the Union" that "there are obviously some real challenges at this point," including a fresh outbreak of violence earlier Sunday in eastern Ukraine.
"But we also believe that there has been some progress. I'm seeing reports this morning that at least one of these government buildings now has a Ukrainian flag flying over it," Pyatt said.