Ibrahim Kalin, chief adviser to Turkey s President Recep Tayyip Erdogan, speaks during an interview in Istanbul. AP
The comments from Turkish President Recep Tayyip Erdogan's top foreign policy adviser followed a round of urgent talks in Brussels that NATO leaders had hoped would pave the way for the Nordic states' formal approval to join the bloc at the Madrid summit.
Ankara has accused Finland and Sweden of providing a safe haven for outlawed Kurdish militants whose decades-long insurgency against the Turkish state has claimed tens of thousands of lives.
Ankara has also demanded the two countries lift their weapons freezes on Turkey.
Erdogan's foreign policy chief Ibrahim Kalin reported no breakthrough at the Brussels talks.
"The Madrid NATO summit is not the deadline, so our negotiations will continue," Kalin told reporters.
"The existence of terrorist organisations must end in those countries. That is what we expect both from Finland and Sweden."
NATO chief Jens Stoltenberg called the meeting "constructive" while conceding that Turkey's "legitimate" concerns had still not been fully addressed.
"Turkey's has legitimate security concerns over terrorism that we need to address," Stoltenberg said.
"So we will continue our talks on Finland and Sweden's applications for NATO membership, and I look forward to finding a way forward as soon as possible."