Luke Mitchell and Nat Fraser and seek to overturn their convictions
Luke Mitchell, 21, was told by appeal judges at the Court of Criminal Appeal, he would be allowed more time to seek legal advice.
But Nat Fraser, 51, heard his latest legal challenge could go no further.
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Hide AdAlthough both have tried to win their freedom, and lost, lawyers thought they had found a new way to question previous court decisions.
Mitchell is serving a minimum of 20 years for a savage knife attack on girlfriend Jodi Jones.
The 14-year-old's body was found in June 2003 in woods beside a path which joined her house in the Easthouses area of Dalkeith with the Mitchell home in the Newbattle district.
Fraser was ordered to serve at least 25 years by judge Lord Mackay of Drumadoon who branded him "evil" and "cold-blooded" for arranging for a hit man to execute estranged wife Arlene, 33.
The latest challenges were presented as petitions to the nobile officium – a sort of last resort process used in Scottish criminal cases where there is no other form of redress.
A final ruling on the appeal will come at a later date.