Russia went glaringly absent from President Donald Trump’s speech to Congress last week. Although Russians expect relations with Washington to change, they are anxiously asking how. Will Mr. Trump confront Moscow with the brash rhetoric he has directed at others? Will he be President Vladimir Putin’s lap dog, as many American critics and Kremlin propagandists predict?

Or will Mr. Trump surprise the world by forging a relationship with Mr. Putin and striking a deal that could help pull my country out of its decline?

Russians wait with both hope and apprehension as they grapple with the uncertainty of Mr. Trump’s message and Mr. Putin’s future. Many expect some kind of deal. In any negotiation, each side must understand the needs and desires of the other. What kind of bargain can Mr. Trump offer, and what can Mr. Putin do in exchange?

Let us break with conventional wisdom for a moment and set aside the expected American preconditions for normalized relations—the difficult matters of Syria, Ukraine, Iran, nuclear weapons and more. These are urgent issues. But no real progress is possible if Americans fail to understand the real, and very different, problems that face Mr. Putin, those around him, and Russia itself.

For today’s regime in Moscow, the overriding goal is to defer the inevitable question of leaving power. Military adventurism, like the police state itself, is a means to that end.

Mr. Putin has been in power for nearly 18 years—like Leonid Brezhnev, whose rule (1964-82) became reviled as the “age of stagnation.” Today, as in the late Brezhnev years, Russia’s economy is languishing and the standard of living is falling. Education, science and health care are decrepit. Russians tell pollsters, whom they assume to be agents of the FSB secret police, that they support the regime. Then they don’t go to vote.

As Brezhnev did in Afghanistan, Mr. Putin has boosted his popularity by creating his own unprovoked war in Ukraine, to make Russians feel that he is defending them from an external threat. The Kremlin insists Mr. Putin’s next presidential term will be his last, and that he would like to retire by 2024. But he cannot do so without taking a huge personal risk.

No Russian leader leaves power willingly. Since the Middle Ages, those who did not die a natural death in office were either assassinated, executed, forced to resign, overthrown or some combination thereof. One slight exception is Boris Yeltsin. In 1999 he transferred power to Mr. Putin, who promptly destroyed any positive legacy Mr. Yeltsin might have had. There were also two pretend departures, in 1560 when Czar Ivan the Terrible left the throne temporarily and put another man in his place before returning, and in 2008 when Mr. Putin installed Dmitry Medvedev as a token president for one term.

Russians today understand that a change of regime is inevitable, and that postponement, especially through Mr. Putin’s methods, only worsens the eventual outcome. He enjoys support not because the people love him or are satisfied with his policies. The people support him because they can’t imagine an alternative.

The regime maintains fear about what comes next by eliminating opponents. Most are simply destroyed politically, but some are physically liquidated. Electoral fraud, repressive laws and constant, paralyzing propaganda reinforce the expectation that the current regime will survive

The situation is an ever-deepening spiral. Russian society is losing its reserves of trust. Nobody can guarantee when today’s leaders will leave, and whether that transition will be peaceful and orderly or violent and bloody. Mr. Putin can’t step aside without such guarantees. He is destroying society to postpone his departure. But a damaged and fearful society is incapable of assuring an orderly departure.

Mr. Trump has an opportunity to begin a businesslike dialogue with Mr. Putin—not about the vital issues in the headlines, but about a far more important problem: how to avoid unnecessary conflicts inside and outside Russia by ensuring a smooth transition of power.

It raises important questions that must be considered now: What does the Kremlin have to do at home and abroad to make this soft landing possible? What reciprocal steps and guarantees can the West offer? It’s time for the world to start contemplating a post-Putin world.

Such guarantees are not merely personal. The dialogue must confront some unpleasant realities. Mr. Putin would like to use international agreements to preserve and formalize the “gains” he has achieved in Europe: his conquest of Crimea and neutral status for Ukraine and other states Mr. Putin considers to be inside Russia’s legitimate sphere of influence.

The desire for such concessions may motivate Mr. Putin to discuss the matter of his stepping down while Mr. Trump is in office. That, in turn, could create an opportunity to remove other problems from the table. But it could also tempt the West to appease and buckle under, or to throw up barriers.

In any other scenario, if Mr. Putin is going to be thinking about how to remain in power, he needs the U.S. for only one thing—to play the role of a “safe enemy.” That allows him to rally the Russian people around him, while knowing America presents no actual danger. To expect any other approach from Mr. Putin is a self-delusion that will carry a high cost in the end.

If concessions offered by Mr. Putin are not to America’s benefit, Washington will need to acknowledge that the only workable policy toward the Kremlin is a Cold War-style containment, with clearly defined parameters.

The window for handling Mr. Putin is very narrow. Mr. Trump’s brashness has stopped at attacking Mr. Putin personally. That atypical restraint horrified many Russia hawks in the West. But it may turn out that Mr. Trump took a better approach. Whether accidentally or by design, he has left the door open for Mr. Putin to make a graceful exit. That would be good for everyone.

Mikhail Khodorkovsky for The Wall Street Journal