NATO doesnt have any leverage on Turkey they can choose non NATO suppliers as long as the alternatives are competitive, Turkey is making attempts to be not dependent on US. In TFX case EJ200 already chosen as the basis for a new power plant.
https://en.wikipedia.org/wiki/TAI_TFX#Propulsion
Turkish Prime Minister
Ahmet Davutoğlu announced on January 8, 2015, that the TFX will be a twin-engined fighter jet.
[2] The Turkish Undersecretariat for Defense Industries (SSM), the procurement agency for
Turkish Armed Forces, has written a letter of intent to three engine manufacturers: General Electric, Pratt & Whitney and EUROJET Turbo.
On 20 January 2015,
Aselsan of Turkey announced that it had executed a memorandum of understanding with Eurojet, the manufacturer of the EJ200 engine used in the Eurofighter Typhoon.
[29] The announcement also stated that a derivative of the EJ200 will be used in the TFX program.
[30][31][32][33] The two companies will additionally collaborate and co-develop engine control software systems and engine maintenance monitoring systems.
[33] Turkey's selection of the EJ200 evidences TAI's intention to utilise supercruise capability.