By all accounts, Mr. Obama has the power to go much further even without new legislation from Congress, which nonetheless is advancing competing bills to punish Russia. He signed an executive order last week authorizing not just additional visa bans but also asset freezes and other financial moves. But he did not actually invoke it against any targets, preserving what he called “the flexibility to adjust our response going forward based on Russia’s actions.”

Administration officials said the internal debate was not whether to penalize Russia for its actions — there is broad consensus for that, one that hardens with each passing day — but when, and how hard. Beyond freezing assets of individuals, the administration could sanction banks and potentially cut the country off from the dollar economy.

“There are people who are cautious about going too far too quickly, and there are those who want the president to show resolve,” said a former administration official in touch with the White House.

According to current and former officials, those most supportive of strong action include officials closest to the situation in Ukraine: Victoria J. Nuland, the assistant secretary of state for the region, and Geoffrey R. Pyatt, the ambassador to the country, as well as Daniel Fried, the State Department coordinator of sanctions. Those more wary about crippling the Russian economy include Jack Lew, the Treasury secretary; Michael Froman, the president’s trade representative; and Caroline Atkinson, the president’s international economics adviser.

The immediate question is whether to take action before Sunday’s referendum in Crimea in hopes of forestalling it or instead to wait and see what Moscow does after it is over. The issue is further complicated by the fact that financial sanctions take time to prepare. While the United States has broad latitude to bar foreigners from traveling here, the government needs to build an evidentiary case that could withstand possible court challenge when it freezes assets.