With the fight for Mosul entering its second week, the Obama administration is under pressure to declare victory over the Islamic State, a win the White House says would validate the president’s strategy of waging proxy wars against the terrorist group in Iraq and Syria days before Americans head to the polls.

A quick victory by U.S.-backed Iraqi and Kurdish forces also would reinforce Iraqi Prime Minister Haider al-Abadi’s tenuous hold on power in Baghdad, where he faces an increasingly frustrated Sunni bloc in parliament and Shiite factions allied with former Prime Minister Nouri al-Maliki looking to undermine the regime.

But wresting total control of Iraq’s second-largest city from the Islamic State — also known as ISIS, ISIL and Daesh — likely will take months. Iraqi forces face heavy fighting by a tenacious enemy as they close in on Mosul.

Obama administration officials may well declare victory in Mosul once pockets of Islamic State resistance are cleared out, but the challenge will be defining a win and making sure it sticks, said Thomas Donnelly, a resident national security fellow at the conservative American Enterprise Institute in Washington.

“For the Obama administration, there is an incentive to say, ‘Yes, our strategy of subsidizing Iraqi proxies and of leading from behind and helping the Iraqi government rebuild its military forces to defeat ISIS has been a success,’” Mr. Donnelly said.

A declared win in Mosul could bolster Democratic presidential nominee Hillary Clinton’s chances to secure the White House, Mr. Donnelly said in an interview with The Washington Times.

“It would be a gift to the Clinton administration,” said Mr. Donnelly, opining that the Obama administration may try to announce a victory in Mosul before Americans go to the polls Nov. 8.

“They certainly don’t need it to win the election [but] there may be some people in the administration with the good sense of not wanting to burden a Clinton administration with the expectation of another Iraq withdrawal,” he said.

An ABC poll has Mrs. Clinton holding a 12-point lead over Republican presidential nominee Donald Trump.

The timing of the Mosul offensive has less to do with the election than with securing President Obama’s foreign policy legacy.

“You definitely have political motivations on the part of both the Iraq and the U.S. governments to move quickly with this campaign,” said Linda Robinson, a senior international policy analyst with the Rand Corp.

“The Obama administration — and it’s no secret by their own statements — has always wanted to see this offensive move forward before the end of the year and thus before the end of its term,” said Ms. Robinson, who was a top civilian adviser to Adm. William McRaven, former U.S. Special Operations Command chief.

That said, “the American people, unfortunately, are not really watching [Mosul], and it’s not going to affect the American elections,” she told The Times.

White House press secretary Josh Earnest acknowledged the watershed moment that the fight for Mosul posed to the president’s counterterrorism doctrine.

“I think the president would be the first to acknowledge that this is a significant test, given the population size of Mosul, given the large geographic area that it encompasses,” he told reporters in Washington when the Mosul operation kicked off Oct. 17.

Heavy reliance on local armies and paramilitary groups, trained and armed by small U.S. special operations teams and backed by American surveillance drones and air power, has been the administration’s modus operandi in battling al Qaeda, the Islamic State and other extremist groups worldwide.

White House and Pentagon officials have cited U.S.-backed operations to defeat al Qaeda’s Yemeni cell, as well as operations in Afghanistan against the Taliban, as clear examples of the strategy’s success.

But the administration’s political opponents, led by Mr. Trump, say deteriorating security in both countries and the failed effort to train Syrian militias are clear repudiations of the Obama counterterrorism doctrine.

During his third and final presidential debate with Mrs. Clinton on Wednesday, Mr. Trump asserted that the White House timed the Mosul operation explicitly to provide maximum political benefit for the Democratic nominee.

“The only reason [the White House] did it is because she’s running for the office of president and they want to look tough,” Mr. Trump said during the debate.

The administration has “made so many mistakes” in the run-up to the Mosul operation, Mr. Trump said regarding the fight against the Islamic State. He said Mrs. Clinton “wanted to look good for the election, so they’re going in.”

However, Mr. al-Abadi’s government, not the White House, made the call to begin the assault.

It is Mr. al-Abadi’s political future, not the next occupant of the White House, that is truly in the balance as Iraqi forces close in on Mosul, Ms. Robinson said.

“He needs that political shot in the arm right now. His government has been very weak, and nothing would shore him up like a successful Mosul push,” she said.

Mr. Donnelly agreed with that assessment but warned that Mr. al-Abadi’s rush to victory in Mosul could pose more of a threat to Iraq than the actual fight to reclaim the city.

“If we declare victory too soon and the American presence is reduced or eliminated entirely, there’s some chance that we’ll soon see a revival of al Qaeda in Iraq, like an al Qaeda in Iraq 3.0,” he said.

The Islamic State is reverting to guerrilla tactics and terrorist strikes even as it digs in for a protracted fight in Mosul.

Islamic State operatives have carried out a growing wave of suicide bombings in the months leading up to the Mosul offensive, targeting Shiite populations in Baghdad and elsewhere in a clear echo of the efforts by the group’s predecessor, al Qaeda in Iraq.

U.S. and coalition military officials estimated Wednesday that they would need 35,000 to 40,000 troops to hold Iraq’s second-largest city once the Islamic State is driven out.

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