An Israeli soldier last month on a street that separates an Israeli settlement and a Palestinian neighborhood inside the West Bank city of Hebron. President Trump’s comments on a two-state solution have prompted confusion in the region. CreditChris McGrath/Getty Images

JERUSALEM — The huge billboard images appeared overnight in Tel Aviv: a menacing crowd of Palestinians making the V for victory sign and bearing a legend in Arabic, “Soon we will be the majority.”

One interpretation of that inevitability was explained in Hebrew for those who dialed the number on the billboard: If Israel does not act to separate itself from the Palestinians, it will be less secure, less democratic and less Jewish. The provocative — many said racist — campaign was kicked off last month by retired Israeli generals and senior officers to shake Israelis out of apathy.

President Trump accomplished something similar over the course of just a few seconds on Wednesday, when, standing beside Prime Minister Benjamin Netanyahu at the White House, he declared that he was “looking at two-state and one-state” formulas for resolving the Israeli-Palestinian conflict.

“I like the one that both parties like,” he added, seemingly overturning decades of American policy centered on the creation of a Palestinian state alongside Israel.

Purposefully or not, Mr. Trump had suddenly implied that the long-proposed solution of two states did not really matter.

By Thursday, Israelis and Palestinians were feverishly debating what might come next, still confused about American policy after Mr. Trump’s ambassador to the United Nations, Nikki R. Haley, reasserted that the administration “absolutely” supported two states.

What were the viable options other than the two-state solution? One state with equal rights for both Israelis and Palestinians? A dominant Israeli state alongside a defined Palestinian region with statelike but curtailed powers? Would either side ever settle for less than everything?

Over decades, Palestinians have watched Jewish settlements spread over land they consider theirs for a future state and concluded that Israel did not intend to concede it. Many of them, particularly those in Hamas, which rules the Gaza Strip, still do not recognize Israel at all.

Many Israelis believe that they have repeatedly made good offers that were refused, and that Palestinians are irrevocably split between the West Bank and Gaza, with no unified leadership to negotiate with. Good will, they say, has been met by rocket fire.

Now, the Israeli political establishment, moving rightward, clearly believes it is the time to put its thumb on the scale.

“I think what the president and prime minister were saying was any solution is possible and now we have to look at alternative solutions, and there are alternative solutions,” said Michael Oren, a deputy minister for diplomacy in Mr. Netanyahu’s office.

These, Mr. Oren told reporters, could involve “interim measures and recognition of the fact that there may be a two-state reality on the ground, which may not conform to what we know as a two-state solution, but would enable the Palestinians to lead their lives in prosperity and security” — and also benefit Israel.

Mr. Netanyahu, weakened by corruption investigations and under pressure from right-wing politicians who oppose a Palestinian state, has recently been evasive about his support for a two-state solution. It depended, he said in Washington, on what the Palestinians had in mind: “What are we talking about? Are we talking about Costa Rica, or are we talking about another Iran?”

Mostly, Mr. Netanyahu appears to want to solidify Israeli control over the occupied West Bank and manage the conflict. That basically means maintaining the current situation of Palestinian cantons divided by growing Israeli settlements and surrounded by Israeli forces.

Mr. Netanyahu has referred to it as a “state-minus” — implying the Palestinians would get some statelike autonomy, and that would be enough. Critics call it a creeping one-state reality, and certainly not the “ultimate deal” that Mr. Trump says he hopes to achieve.

Some analysts chalk up Mr. Trump’s flippancy to a lack of knowledge, because one thing many Palestinians and Israelis do agree on is that a one-state formula will not bring peace.

“One state is not an option,” said Ghassan Khatib, a Palestinian political scientist at Birzeit University in the West Bank, noting that Israel, which was established to give Jews self-determination, would never give all Palestinians the vote. “We are talking two states or no solution, a continuation of the status quo,” he said.

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A Palestinian worker building a new house in an Israeli settlement in the West Bank. Over decades, Palestinians have watched Jewish settlements spread over land they consider theirs for a future state.CreditChris Mcgrath/Getty Images