Live outdoor events will be restricted to 500 spectators in Scotland from Boxing Day onwards.
The decision was announced by First Minister Nicola Sturgeon following a Cabinet meeting which discussed ways to slow the spread of the Omicron variant of Covid-19.
The new restrictions will be in place for “a period up to three weeks”.
Glasgow and Edinburgh are due to meet in a festive double header, with the first leg scheduled for Scotstoun on December 27 and the return in the capital on January 2.
The matches will go ahead but neither club is likely to take up the offer of admitting 500 supporters.
It is not financially viable to admit such a small number given the cost of stewarding and other associated match-day costs.
A statement from Edinburgh said: “Following today’s Scottish Government announcement, Edinburgh Rugby’s 1872 Cup derby in January will regrettably be spectator free.
“The club will contact all season ticket members and ticket purchasers in due course with a further update.
“We thank supporters for their patience and understanding, and encourage all to stay safe during the festive period.”
Glasgow Warriors said: “Following the Scottish Government's latest announcement, Glasgow Warriors’ upcoming fixture against Edinburgh (on Monday 27 December) will be played to all intents and purposes spectator-free.
“We’re incredibly disappointed to not have the Warrior Nation at Scotstoun Stadium with us. Members and ticket purchasers will be contacted in due course. The club thanks you for your patience at this time.”