The firm, which bought Morrison for £262m in 2000, is trying to recoup £100m on the acquisition.
It has been reported that Sir Fraser and other Morrison directors will receive letters from AWG's lawyers asking them to reach an out-of-court settlement or be sued. It is understood that Sir Fraser's brother, Gordon Morrison, who was an AWG board member until last May, is not involved in the dispute.
Since buying Morrison, AWG has been hit with losses of £106m from six poorly performing Morrison contracts. These include projects in Manchester and Birmingham. The utilities group has been threatening legal action since the losses first came to light in May last year.
AWG has been under pressure from shareholders to try to recover the money after the group posted a pre-tax loss of £53m for the year to 31 March. AWG held its annual general meeting on Wednesday.
Sir Fraser could not be reached for comment but it is understood he will contest any legal action.
He is currently chairman of Edinburgh-based architect RMJM in which he took a stake after a management buyout in April.