Last year, Webster, who has always maintained his innocence, was granted leave to appeal his murder conviction and sentence, which he says was too harsh.
A Court of Criminal Appeal hearing began in Edinburgh yesterday.
BBC Scotland reports defence lawyers argued there had been insufficient evidence at trial to prove that 32-year-old Ms Morris' death had not been a tragic accident.
The prosecution will present answers to the defence claims tomorrow, with a decision by judges Lord Eassie, Lady Clark of Calton and Lord Wheatley to follow.
A 2011 trial heard how Webster drugged his first wife Ms Morris just eight months after their marriage and drove his car off the road with the unconscious woman inside.
He then torched the vehicle and covered his tracks before collecting the life insurance payout and moving to New Zealand.
The jury then heard Webster married Ms Drumm before attempting a similar deadly scam in 1999 near Auckland by drugging her and planning to kill her in another staged smash.
On his return to Scotland, he tried to con another woman into a bigamous marriage, and even getting her to change her will to leave him everything, including her house and a luxury yacht.
