Ashraf's resignation offer was announced by a private television channel Geo but no confirmation was immediately available from a board official.
The channel said Ashraf had sent his resignation to the chief patron of the board, President Pervez Musharraf, but it was not known if he had been accepted.
Ashraf who took over as chairman six months ago had earlier in the day told reporters that he was sorry for the performance of the team and could only apologise to them.
Pakistan, the 1992 winners, crashed out of the competition on Saturday, losing their second match to debutants Ireland by three wickets.
Less than 24 hours later, their coach Bob Woolmer died after being found unconscious in his Kingston hotel room. The cause of death has yet to be announced by Jamaican authorities.
Later on Sunday, their World Cup captain Inzamam-ul-Haq announced he would be quitting one-day international cricket after the competition and stepping down as Test skipper.
The seven-week World Cup culminates in the final in Bridgetown, Barbados on April 28.