Aaron Campbell took the six-year-old from her bed at her grandparents' home on the Isle of Bute on July 2 last year.
Earlier this year, the now 17-year-old Campbell was found guilty at the High Court in Glasgow and handed a life sentence with a minimum of 27 years.
His appeal is expected to be heard before three judges at the Criminal Appeal Court in Edinburgh.
During the nine-day trial in March, he told the jury about how he would "never do something like that" when asked if he had killed Alesha.
But a friend had shown detectives messages he sent in 2017, saying he "might kill 1 day for the lifetime experience".
Campbell also tried to convince the jury he had sex with Toni McLachlan, the girlfriend of Alesha's father Robert MacPhail, on the night of the murder.
After carrying out the atrocity, Campbell exchanged messages with Ms McLachlan, who he later tried to blame for the crime, saying of the missing girl: "Oh damn am sure she's not went too far."
He later confessed the crime to those assessing him ahead of the sentencing, saying he was "quite satisfied by the murder".
Alesha, from Airdrie, North Lanarkshire, had been staying with her family on the island during the summer holidays.
Following the discovery of her body it was Campbell's mother who called police in an apparent bid to eliminate her son from their inquiries.