Jonathan Nelms, a Baker McKenzie partner in Washington, D.C., who spent nearly three years toiling in the firm's Moscow branch, chooses his words carefully when describing the current state of relations between this country and Russia.

“I don't want to call it a new Cold War,” said Nelms, whose firm has 120 lawyers in Moscow and St. Petersburg.

But the relationship has chilled enough to have business consequences for U.S.-based firms that have Russian outposts, according to Nelms and other partners at five different firms—most of whom spoke only on condition of anonymity.