The Islamic community itself had previously classified this allegedly right-wing extremist mosque as illegal.
In several cases the process of expelling imams connected to the Turkish-Islamic Cultural Associations (ATIB) organisation was underway, Kickl said.
Cavusoglu, in a phone call with his Austrian counterpart Karin Kneissl, expressed his "discomfort" over Kurz's statement on the issue, the report said.
"This is just the beginning", far-right Vice Chancellor Heinz-Christian Strache said at a news conference Friday.
"If these measures aren't enough, we will if necessary evaluate the legal situation here or there", he said.
Speaking at the news conference, Interior Minister Herbert Kickl said the authorities were reviewing the residence permits of about 40 imams employed by ATIB, and their family members, over concerns they were being paid from overseas.
The clampdown comes after Austria's religious affairs authority investigated images published in April of children in a Turkish-backed mosque playing dead and reenacting the World War I battle of Gallipoli.
Kurz, 31, became chancellor of Austria in December after his conservative People's Party entered a coalition with the far-right Freedom party.
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Among Austria's 8.8 million people, are an estimated 600,000 Muslims.
Kurz pointed out that up to 60 imams may be deported, or may be prevented from obtaining visas on the pretext of receiving external funding, stressing that associations that represent political Islam and extremist tendencies have no place in Austria.
"It is an attempt to target Muslim communities for the sake of scoring cheap political points", spokesman Ibrahim Kalin wrote on Twitter. The move invokes a 2015 law banning religious groups from getting foreign funding, and the government suspects that as many as 60 imams may need to leave the country.
They are then shown to play dead with Turkish flags draped over their "bodies". Two had already had their permits revoked, while five more were denied first-time permits.
The move was "a reflection of the Islamophobic, racist and discriminatory wave in this country", Kalin said. The government recently announced plans to ban girls in elementary schools and kindergartens from wearing headscarves, adding to restrictions on veils.
Imams in Austria gather in front of a mosque in Vienna on June 14, 2017, expressing their solidarity against extremism.
Last week Turkish President Recep Tayyip Erdogan attacked Mr Kurz, saying: "This immoral chancellor has a problem with us".