President Biden affirmed his support for the sale of a new fleet of F-16 jet fighters to Turkey on Thursday, two days after the Turkish president dropped his threatened veto of Sweden and Finland’s membership in the North Atlantic Treaty Organization.

“We should sell them the F-16 jets and modernize those jets as well. It’s not in our interest not to do that,” Mr. Biden said at a news conference after the NATO leaders’ summit in Madrid.

“There’s no quid pro quo with that, it’s just that we should sell. But I need congressional approval to be able to do that, and I think we can get that,” he said.

The comments are Mr. Biden’s strongest signal yet of the administration’s support for the proposed sale of new F-16s to Turkey, which officials from both countries view as a way to keep the Turkish government in the Western camp opposed to Russia amid its war in Ukraine.

Mr. Biden said he communicated his position on the arms deal to Turkey’s President Recep Tayyip Erdogan during a bilateral meeting on Wednesday, which came after Turkey agreed to allow Finland and Sweden into NATO, dropping objections that threatened to derail an expansion of the alliance intended to counter Russia’s invasion of Ukraine. He said he previously told Mr. Erdogan of his support for the sale, a position that officials later said he communicated to his Turkish counterpart privately in a phone call.

Mr. Erdogan also said Mr. Biden had offered him assurances on the F-16 deal, though the Turkish president didn’t offer specifics of their conversation.

“Mr. Biden said he would take the necessary steps. As you know there are not just Democrats in the Congress, there are also Republicans,” he said.

Mr. Erdogan said a Turkish delegation will travel to the U.S. to seek the support of lawmakers.

Turkey requested the new F-16s from the U.S. last year, with both governments viewing the deal as a way to repair the battered defense relationship between the NATO allies. The U.S. imposed sanctions on Turkey over its 2017 purchase of Russia’s S-400 air-defense system.

The U.S. also expelled Turkey from the advanced F-35 fighter program, saying that Russia might be able to use the S-400 to hack the aircraft’s electronics.

Turkish President Recep Tayyip Erdogan and President Biden met in Madrid Wednesday.

Photo: Susan Walsh/Associated Press

The administration still has yet to send the F-16 proposal to Congress, where it is expected to face some resistance despite Turkey’s decision to drop its veto on Finland and Sweden joining NATO.

The administration earlier this year asked Congress to approve a separate package of Sidewinder and Amraam missiles, radar, and other equipment for Turkey’s existing F-16 fleet. That request is still facing “stiff headwinds” from lawmakers, said a senior congressional aide.

Lawmakers are preparing to attach a series of amendments to the upcoming National Defense Authorization Act, an annual military-spending measure, to express concerns and place conditions on the sale of the missiles in a preview of expected congressional opposition to the F-16 sale, the senior aide said.

Mr. Erdogan’s recent threats to launch a new military incursion in northern Syria have deepened lawmakers’ concerns about further arms sales for the Turkish military, the aide also said.

U.S. officials said the White House never offered the F-16 deal or other aspects of the U.S.-Turkey relationship as a quid pro quo for accepting Sweden and Finland into NATO. In a phone call with Mr. Erdogan on Tuesday, Mr. Biden did say that getting the NATO expansion deal done would open the door to an in-person meeting with him, U.S. officials said.

A senior U.S. official said now that Mr. Erdogan has lifted his ban on NATO accession, it will make the F-16 sale to Turkey an easier sell in Congress.

Mr. Erdogan also received the attention and the public pledges he wanted for a collective crackdown on the Kurdistan Workers’ Party, which is labeled a terrorist organization by the U.S., the European Union and Turkey, another senior official said. Turkey initially accused Sweden of engaging with the group. However, countries in the alliance also shared the view that any missteps could be deemed a win for Russian President Vladimir Putin, and so getting Turkey onboard for Finland and Sweden’s ascension was a priority, the official added.

Turkey, Sweden and Finland signed an agreement on Tuesday in which the two Nordic countries promised not to support Kurdish militant groups and end restrictions on arms sales to Turkey, in return for Mr. Erdgoan’s backing for their accession to NATO. The deal, which came as a surprise to some officials, followed weeks of negotiations that culminated with a four-hour meeting among the leaders of the three countries and NATO officials in Madrid.

Turkey has sought to leverage its role in the Ukraine crisis to seek concessions from both allies and opponents. Mr. Erdogan’s government has facilitated two rounds of Russia-Ukraine peace talks while also selling armed drones that Ukraine has used in its resistance to Russia’s invasion.

In recent weeks the Turkish government has taken an increasingly neutral stance on the Russia-Ukraine war, saying it must be cautious about future arms sales to Ukraine and saying its priority is negotiating a peace deal.

Turkey faces strong opposition in Congress, particularly among some Democrats who regard Mr. Erdogan as an autocrat. Sen. Bob Menendez (D., N.J.)—the chairman of the Senate Foreign Relations Committee and a supporter of Greece, Turkey’s rival—has strongly criticized Mr. Erdogan over his human-rights record and called for withholding weapons sales as a way of pressuring foreign governments.

—Noemie Bisserbe in Madrid contributed to this article.

Write to Jared Malsin at jared.malsin@wsj.com and Vivian Salama at vivian.salama@wsj.com