Hungary has indicated that it will follow Türkiye's lead in the dispute.
Hungarian Foreign Minister Peter Szijjarto said Tuesday he had been holding regular consultations with Turkish FM Hakan Fidan about Sweden.
"In the coming days as well communication will continue to be close and continuous with the Turkish foreign minister," Szijjarto said.
"And if there is a shift, we will of course keep our promise that Hungary will not delay any country in terms of membership."
The Swedish government on Sunday condemned last week's Koran burning as "Islamophobic".
But it added in a foreign ministry statement that Sweden had a "constitutionally protected right to freedom of assembly, expression and demonstration".
The Stockholm police ruled prior to last week's protest that the risks associated with the Koran burning "were not of a nature that could justify, under current laws, a decision to reject the request".
Sweden and its neighbour Finland dropped decades of military non-alignment and applied to join NATO in the wake of Russia's invasion of Ukraine.
Finland formally joined the bloc in April.
