Al-Qabas daily cited Nayef al-Ajmi as saying he will remain in the post after a request from the "political leadership" following a meeting yesterday.
The term "political leadership" generally refers to the ruler of the oil-rich Gulf state.
"I will obey the orders of the political leadership and will continue along the same path I have started," Ajmi, who is also justice minister, said.
Ajmi, who strongly denied the US accusations, said on Friday he had tendered his resignation, citing health problems.
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His appointment as minister in January was a "step in the wrong direction", Cohen said in a lecture in the United States, parts of which were carried by the Kuwait press last month.
Ajmi said on Friday he was resigning because of health problems predating the US accusation.
He said he had been undergoing tests in London when the reports of Cohen's comments surfaced in the Kuwaiti media, and had cut the tests short to head home.
Ajmi acknowledged he had taken part in fund-raising campaigns for Syria, but insisted they had been for humanitarian purposes and not for Syria's Al-Qaeda affiliate Al-Nusra Front as alleged by Cohen.