Webster, who has always maintained his innocence, lost an appeal against his conviction in December.
Judges at the Court of Appeal in Edinburgh rejected a claim that he was the victim of a miscarriage of justice.
Now, three judges will consider an appeal against the length of Webster's sentence at the Court of Appeal on March 14, The Herald newspaper in Glasgow reports.
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 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.