And, at the same time, he is preparing to appeal against a move to ban him from nursing.
Confirmation of the moves comes just two months since the 32-year-old Scots-born nurse was told he must serve a minimum of 30 years in prison for the murder of four patients and the attempted murder of a fifth, at Leeds's two main hospitals.
Norris, described by the trial judge as a "thoroughly evil and dangerous man", killed his victims with insulin overdoses between May and November 2002.
In March this year, he was found guilty after a four month trial at Newcastle Crown Court of four murders and one charge of attempted murder.
The jury found Norris, formerly of Kirkstall, Leeds, guilty of the murders of Doris Ludlam, 80, of Pudsey, Bridget Bourke, 88, of Holbeck, and Ethel Hall, 86, from Calverley, all at Leeds General Infirmary; and Irene Crooks, 79, of Leeds, at St James's Hospital , and the attempted murder of Vera Wilby, 90, of Rawdon, at Leeds General Infirmary.
A spokeswoman for the Court of Appeal in London today confirmed that Norris had applied for leave to appeal against his conviction.
In 2003, Glaswegian-born Norris was suspended by the Nursing and Midwifery Council from practising as a nurse while he was under investigation over the deaths.
A spokeswoman for the Nursing and Midwifery Council, which suspended him in 2003, said today that a hearing was expected within the next few months to formally decide to strike Norris from the Council's register. She added that they had not, at this stage, received any notification of an appeal. If if an appeal was to be held against his criminal conviction, the Nursing Council would still hold disciplinary proceedings.
Last night, Stuart Hall, whose mother Ethel was Norris's fifth victim, told the YEP: "I have no doubts about Norris's guilt. Personally, I half-expected him to appeal because of his arrogance.
"Obviously, the appeal is going to cause distress to the families of his victims again. Like them, I had hoped he would just shut up, take his punishment and let the families of his victims get on with their lives".