WASHINGTON—The Trump administration is discussing whether to press the Afghan government to suspend coming presidential elections, according to people briefed on the discussions, as the U.S. seeks to engage the Taliban in talks to end the 17-year war.

The possibility of such a step, one of several options being considered by U.S. officials, is a sign of the urgency the administration sees in trying to broker a political breakthrough in a conflict that has bedeviled three successive American presidents, according to these people.

To urge a suspension of the April election, an idea these people said was raised by U.S. special envoy Zalmay Khalilzad in talks with various stakeholders and intermediaries, would be a contentious move after the U.S. has long promoted democracy in Afghanistan. Mr. Khalilzad’s office declined to comment on his efforts.

The idea would be opposed by a key player: Afghan President Ashraf Ghani, who is expected to seek a second five-year term in the election. “Continuity in a democratic process is a must, and any other proposal than the will of Afghans, which is outlined in our constitution, is simply not acceptable,” said Mr. Ghani’s spokesman, Haroon Chakhansoori.

Discussing recent parliamentary elections last week in Kabul, Mr. Ghani emphasized that the presidential vote would proceed as scheduled on April 20. “Based on the free election, complete inclusive election, people will decide on their future leadership,” he said.

Pressing Afghanistan to postpone elections would likely create a rift between Washington and Kabul. Because the Afghan government needs U.S. financial and military support from the U.S., the Trump administration has significant sway, but the ultimate decision would be made by Afghan leaders who are wary of undue American influence.

The justification for discussing a possible suspension involves elusive peace talks with the insurgent Taliban. Some American officials fear that the irregularities and protracted political turmoil that routinely accompany Afghan voting could paralyze or destroy any peace process that Mr. Khalilzad succeeds in launching.

U.S. officials are considering other ways to quickly bring the Taliban into a political process with Afghan leaders in Kabul, but they have yet to find enough common ground among competing parties to move the process forward, these people said. Neither the Taliban’s spokesman nor its political representatives in Qatar could be reached for comment.

President Trump has questioned America’s role in Afghanistan, while reluctantly embracing a new strategy last year that increased U.S. troop levels in the country to 15,000. Mr. Khalilzad has told colleagues that he probably has six to 12 months to produce a breakthrough for the president, people briefed on the discussions said.

Mr. Khalilzad is a “man in a hurry,” said Robin Raphel, a former American diplomat who worked for years in South Asia.

The idea of a postponement has received a frosty reception in Kabul, where Mr. Khalilzad met Sunday with Afghan officials and politicians.

Many Afghan officials have also expressed opposition to any suspension, but the plan has quiet backing from some in Mr. Ghani’s government. One senior Afghan political figure said he wouldn’t be opposed to calling off the ballot if there was a chance of deal.

“For a peace settlement, you have to be flexible,” the political figure said. “The (presidential) election is not bigger than peace.”

Another alternative being discussed would allow the election to go forward with the understanding that the new government would serve on an interim basis while the warring parties try to create a governing coalition that would include the Taliban, according to a Western diplomat.

“There is no easy scenario,” the diplomat said.

Another approach would include creation of a special assembly of Afghanistan leaders that would choose a new interim government to run the country while the warring parties work on a plan to end the war. Afghan power brokers, including former Afghan President Hamid Karzai, have been pressing for such an assembly, known as a Loya Jirga.

The Trump administration quietly launched direct talks with the Taliban earlier this year as U.S. military efforts to blunt insurgent advances faltered. Mr. Trump’s year-old strategy for ending the war has produced few positive results.

Afghan government control of districts across the country has fallen to its lowest level since the U.S. Special Inspector General for Afghanistan Reconstruction began assessing the information in 2015. Mr. Ghani’s government depends on the U.S. and other foreign donors for the bulk of its funding, including billions of dollars each year to train and equip its army and police.

Last month’s parliamentary elections were marred by Taliban violence, delays and voting irregularities that raised questions about the legitimacy of the process and Mr. Ghani’s ability to oversee a viable presidential election this spring, when the annual fighting season traditionally gets under way.

That has raised concerns among some Americans and Afghan leaders who worry that Mr. Trump will abandon his South Asia strategy as he prepares to run for a second term.

“I don’t think Trump wants to go to reelection with more troops in Afghanistan than when Obama left office, and no idea what they’re doing there,” said a longtime Afghanistan political observer based in Kabul.

The sense of urgency has helped galvanize efforts by Mr. Khalilzad, a former U.S. ambassador to Afghanistan and Iraq who took the new job in September.

Mr. Khalilzad is expected to meet next week with Taliban leaders in Doha, Qatar, according to people briefed on his travel plans. He may also speak with Taliban members in Dubai, according to an official briefed on the discussions.

The Taliban’s political commission in Doha has been bolstered by the additions of senior figures once held at Guantanamo Bay who could help sway the group’s decision makers based in Afghanistan and Pakistan, Western officials said.

This so-called “Taliban Five” were freed in the 2015 exchange for Bowe Bergdahl, a U.S. Army sergeant who was being held by militants. The Taliban Five included the former head of the Taliban’s army, Mohammad Fazl, who could help carry the group’s dominant military wing into the peace process.

Obstacles to talks include the scale and status of the future American military presence in Afghanistan, and the Taliban’s refusal so far to engage with the Mr. Ghani’s government. Mr. Ghani has offered to enter into talks with the Taliban without conditions, but the Taliban have repeatedly described his administration as a U.S. puppet and refused.

As an inducement to the Taliban to begin substantive talks, Mr. Khalilzad “has suggested that a postponement of the elections could be a component of the peace process,” said a former U.S. administration official briefed on the evolving talks.

In connection with an election postponement, one option being discussed is a conference similar to that held in Bonn, Germany in 2001 that cobbled together a post-Taliban administration in Kabul. This time, however, the Taliban would be represented at the table. Mr. Khalilzad played a key role in the 2001 Bonn conference.

While Mr. Khalilzad is exploring the option of suspending the election, other U.S. officials are working with Mr. Ghani to get ready for the vote in April.

“Our focus is on the presidential election, and we’re getting prepared for them,” said a senior U.S. official involved in Afghanistan affairs.