BRUSSELS—The North Atlantic Treaty Organization is considering appointing a senior official to oversee counterterrorism efforts, a move aimed at meeting one of President Donald Trump’s demands that the alliance focus more on terror threats.

The proposal is similar to NATO’s recent decision to create a top intelligence post, a move that Mr. Trump has repeatedly praised and that he has cited as evidence the alliance has responded to his criticisms and is no longer obsolete.

While no NATO country has vocally opposed the idea of a senior counterterrorism coordinator, some diplomats are skeptical about the role’s impact unless alliance members also agree to expand the organization’s counterterror efforts, including funding additional training initiatives.

NATO diplomats have been discussing how they can expand counterterrorism training, including ways to use allied special operations forces to better train antiterror commandos in the Middle East and Africa. Those proposals could include expanding the work or mandate of the NATO Special Operations Headquarters, which develops NATO counterterrorism plans.
But expanding the alliance’s use of its scarce resources, such as special-operations forces, is difficult and could weigh on NATO’s budget, which some countries oppose expanding.

Bruno Lété, a security expert at the Brussels office of the German Marshall Fund, said the U.S. has indicated it wants NATO to do more to combat terrorism. “NATO allies are going to need to subscribe to Trump’s desire for a new NATO that can engage in counterterrorism efforts,” Mr. Lété said.

Allied ambassadors are set to formally discuss the counterterrorism post and other proposals at a May 5 meeting, officials said. Diplomats have been debating various proposals as they prepare for the meeting of allied leaders, including Mr. Trump, later this month. Turkish, British and French delegations have circulated papers.

The U.S. however hasn’t submitted a paper or made any formal requests to the alliance. While Mr. Trump has said he wants the allies to do more on counterterrorism, neither he nor other U.S. officials have stated any specific desires, according to allied diplomats.

Some NATO allies have said privately that without a formal proposal from the U.S., reaching consensus on new counterterrorism plans is difficult.

Some allied military officers have said current training efforts are disjointed and could be better coordinated. While there is currently a midlevel official assigned to coordinating NATO’s various counterterrorism efforts, some officers say a high-level official could expand alliance and national training initiatives.

German diplomat Arndt Freytag von Loringhoven leads the new intelligence department that was formed by merging old functions. He was given the title of assistant secretary-general and a mandate to increase allied intelligence-sharing. Several allies have praised the office’s early efforts.

NATO diplomats said a new department devoted to counterterrorism isn’t under consideration. Instead, the counterterrorism portfolio would be given to an existing NATO official, likely Sorin Ducaru, a Romanian diplomat who is assistant secretary-general for emerging security challenges.

More broadly, other diplomats question whether a larger role for NATO in fighting terrorism is wise, or if money would be better spent on national efforts.

“We have trouble identifying what more NATO could do,” said one European diplomat. “NATO is not the relevant body to do more fighting terrorism.”

One senior diplomat said that while NATO will never be “the first responder,” it can do more to combat terrorism and should continue expanding intelligence sharing on terror threats, such as providing more information about internal threats that could impact the wider alliance.

Some countries have revived the idea of NATO formally joining the coalition against Islamic State.

Germany and Italy remain skeptical of the proposal, alliance diplomats said. Officials from those countries have long argued it could put a Western focus on an effort that must encompass Middle Eastern partners. Other diplomats said they could support the idea if the U.S. makes clear that the step is important. Officials say it is unlikely to make a significant difference to the anti-Islamic State coalition.

U.S. officials in Washington have been debating how hard to push NATO to formally join the counter-Islamic State fight, according to people briefed on the discussions.