Webster, who has always maintained his innocence, lost an appeal against his conviction in December. In February, he launched an appeal against the length of his sentence. It was later dropped.
He has now asked the Scottish Criminal Cases Review Commission (SCCRC) to examine his conviction and sentence, The Herald Scotland reported.
The commission's role is to review and investigate cases where it is alleged that a miscarriage of justice may have occurred in relation to conviction, sentence or both, the newspaper said.
A 2011 trial heard how Webster drugged his first wife Ms Morris, 32, just eight months after their marriage and drove his car off the road with her unconscious inside.
He then torched the vehicle and covered his tracks before collecting the 200,000 pound ($390,600) 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. He stood to gain 750,000 pound ($1.4 million) in insurance money.
On his return to Scotland, he tried to con another woman into a bigamous marriage, even getting her to change her will to leave him everything, including her house and a luxury yacht.