Waiter Surjit Singh Chhokar died following an incident outside his girlfriend's house in Overtown, Lanarkshire, on November 4, 1998.
The 32-year-old father-of-two suffered a fatal stab wound to the heart. Three men were cleared of murder after two separate trials.
Prosecutors want to use a change in the law which opens the way for a possible re-trial.
The Double Jeopardy (Scotland) Act 2011 allows the Lord Advocate to ask appeal judges to "set aside" the acquittals.
No details of the hearing can be reported, for legal reasons.
David Montgomery, 37, Ronnie Coulter, 46, and Andrew Coulter, 33, must wait for further court hearings before the judges announce their decision.