The US and Turkey have released opposing accounts of a phone call between Donald Trump and Recep Tayyip Erdoğan, highlighting deepening tensions between the two countries amid a Turkish military campaign inside Syria last weekend.

According to the White House readout of a call between the two leaders, Trump relayed “concerns” over the escalating violence in the Kurdish enclave in Afrin and urged Turkey to “de-escalate, limit its military actions and avoid civilian casualties”. Trump also called on Ankara to avoid actions that risked conflict with US forces, which have provided arms and air cover for the Kurdish militias Turkey is now battling in Syria, the White House said.

However, a Turkish official said the White House readout did not accurately reflect the content of the call. Trump did not raise concerns about escalating violence in Afrin, the official said, and the two presidents had simply exchanged views on the operation.

The official also denied that Trump had “expressed concern about destructive and false rhetoric coming from Turkey”, as mentioned in the White House readout, saying Trump had instead stressed that open criticism of the US in Ankara was raising concerns in Washington. The official said Erdoğan replied that US policies – such as support for the People’s Protection Units (YPG) and the “harbouring” of Fethullah Gülen, an exiled cleric accused of masterminding the 2016 attempted coup – had “caused outrage among the Turkish people”.

The differing accounts offer a glimpse into how deep the mistrust has become between the two largest military powers in Nato, over the issue of American support for the Syrian Kurdish fighters.

Tensions between Ankara and Washington have long simmered over the latter’s backing of the YPG, a Kurdish militia that has spearheaded the ground campaign of the US-led coalition against Isis in Syria. Turkey considers the YPG to be the Syrian arm of the Kurdistan Workers’ party (PKK), a designated terror group that has fought a decades-long insurgency against the Turkish state.

Those tensions boiled over after the US announced earlier this month that it would build a 30,000-strong border force, including the YPG, that would patrol Syria’s frontiers – a situation Turkey said constituted an intolerable national security threat.

Erdoğan has recently stepped up his rhetoric against the US, accusing it of building a terror army across the border. Relations between the two men have deteriorated after a strong start, in which Trump expressed his respect for Erdoğan after the defeat of the attempted coup and the Turkish president praised his counterpart’s “legendary” election victory.

In further comments suggesting Turkey could expand the offensive in Afrin to other areas, the Turkish prime minister, Binali Yıldırım, has said his government would not allow the creation of a “terrorist” entity along Turkey’s borders.

In a speech in Ankara on Thursday, Yıldırım said: “It is astounding and unacceptable ... that a country which is supposed to protect Nato’s borders is giving open support to armed entities that target our borders.”

Yıldırım said as many as 300 militants had been “neutralized”, killed, injured or taken captive so far in the Afrin offensive.