The National Accountability Court had indicted Dar on September 27 in a corruption case but has continued to serve as finance minister amid strong criticism from opposition parties.
Different media channels, including Geo TV reported citing sources that 67-year-old Dar has sent resignation to Abbasi, forcing the government to clarify the confusion.
Mussadik Malik, PM House spokesman, said that Dar has not resigned but under rules his ministry was run by the Prime Minister.
Earlier, the Dawn reported that government has decided to replace Dar who is facing a corruption case and also away from country to seek medical treatment in UK.
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"The decision was taken after getting a green signal from former prime minister Nawaz Sharif," a senior government official told Dawn.
The source said a list of leaders to replace Dar had already been drafted, the report said.
Two of the people being considered for the job are Shaukat Tarin, who served as finance minister in the Pakistan Peoples Party (PPP) government and Ishrat Husain, who was governor of the State Bank of Pakistan during Pervez Musharraf's regime.
Dar is accused of accumulating assets worth an estimated Rs 831.7 million, which is disproportionate with his known sources of income. The accountability court in Islamabad has already issued non-bailable arrest warrant against him.