The move comes two months after Mr Ashraf’s predecessor Yousuf Raza Gilani was found guilty of contempt over the same issue and barred from office.
There has been a long-running standoff between the government and the judiciary in Pakistan.
Observers fear the tussle could lead to further political instability.
Judge Asif Saeed Khosa said that Mr Ashraf had been given notice “under [the] contempt of court act 2003, read with article 204 of the constitution to show cause as to why he may not be proceeded [against] in contempt of court by not complying [with the] relevant direction of the court”.
“He shall appear in person at the next date of hearing. Hearing adjourned until 27 August,” Mr Khosa said.
The judge said that it was unfortunate that the prime minister had failed to comply with its orders. He warned that Mr Ashraf could be charged with contempt of court.
The charges against Mr Zardari date back to the 1990s when his late wife, Benazir Bhutto, was prime minister.
He and Ms Bhutto were accused of using Swiss bank accounts to launder bribe money. Mr Zardari insists the charges are politically motivated.
Government lawyers insisted then – as they do now – that the president enjoys immunity from prosecution as head of state.