One remaining concern for some lawmakers is how Turkey might use any U.S.-provided fighters. Greece accused Turkey of violating its airspace more than 10,000 times last year, and Turkey has threatened areas in Syria held by Kurds the U.S. considers allies.

Senate Foreign Relations Chair Bob Menendez (D-N.J.), who had previously blocked the fighter transfer to Turkey, said he is in talks with the Biden administration about his hold and that he could make a decision “in the next week,” suggesting he could lift it. He said his longstanding objection over Turkey’s aggression toward Greece and Cyprus still needs to be addressed.

“We’re having conversations with the administration. I still have concerns. I’ve told them about those concerns. If they can find a way to ensure that Turkey’s aggression against its neighbors ceases — which has been great, but that has to be a permanent reality,” he said.

Menendez wants assurances “if there is to be any sales to Turkey, that they will not use them to act in the belligerent way they have against other NATO allies, not just Greece,” he said.

Those guarantees would have to be backstopped by hard power. If the administration beefs up Greece’s military such that it gains a “qualitative military edge” over Turkey, Menendez said “there may be a pathway” to approving the jets.

One element would be to agree to Greece’s request to buy advanced F-35 warplanes, he said, adding that the administration supports that sale but has not formally notified Congress of an approval.

A spokesperson for the State Department declined to comment on any sales before Congress is formally notified.

Senate Foreign Relations Committee ranking member Jim Risch (R-Idaho) said that, like Menendez, he’s still in talks with the administration about ending Turkey’s belligerence against its neighbors. Monday’s breakthrough on Sweden, to Risch, means talks on the F-16s can begin.

“That issue is resolved now. It’s time to talk about the F-16s, and there’s a path forward there,” Risch said. “I think there’s a lot of moving parts here. Having said that, I don’t think there’s anything that’s not resolvable.”

House Foreign Affairs Chair Mike McCaul (R-Texas), and ranking member Gregory Meeks (D-N.Y.), who also have the power to block U.S. arms sales abroad, said they have concerns that must be met before they would approve the F-16 sales.

McCaul has said he could be in favor of upgrades for Turkey’s existing F-16s if Greece receives the F-35. On Tuesday, he said in an interview that Ankara also needs to start talks with Athens, press Russia to extend a Black Sea grain deal and take other steps.

“I’m optimistic,” McCaul said of Turkey getting approvals to buy F-16s. “The first step was letting Sweden into NATO, and they’ve done that now. The rest are sort of sideline issues.”

Meeks said Tuesday on CNN that his hold remains in effect. While the Sweden breakthrough was a “giant step forward,” Turkey can’t resume its violations of Greek airspace.

“But so far so good. I’m seeing signs over the last several months that Turkey has kept its word as far as intrusions and flyovers of its neighbors,” Meeks said. “I need to have further dialogue and conversation but I think that things are moving along quite quickly and quite well.”

While the four committee leads have the power to block the sale, Senate Foreign Relations Committee member Chris Van Hollen (D-Md.) has been an influential opponent of F-16s for Turkey regardless of whether it relents on Sweden.

On Tuesday, he said he hadn’t changed his mind yet and that Ankara would at least have to pledge not to use them against NATO members or America’s Kurdish allies.

“We’re gonna have to look at all this as part of the mix going forward, but I do want to say it’s a big step forward, obviously, that Turkey finally agreed that Sweden could be part of NATO,” he said.

Lara Seligman contributed to this report.