The sides were due to meet at Scotstoun on Monday night but the game was called off in the afternoon due to “a number of positive cases”.
A decision on the return match in the capital will be made later this week as both clubs continue to test their players regularly.
The postponement creates a fixture headache for the United Rugby Championship, with no obvious free dates in a congested calendar.
Both clubs are keen to avoid playing the game during the Six Nations when they would be without large numbers of Scotland internationals whose absence would strip the fixture of much of its lustre.
Other potential new dates could include the weekends set aside for the European competition knockout rounds in April and May but that would hinge on the Scottish clubs’ progress in their respective tournaments.
Both Edinburgh and Glasgow are due to play URC fixtures in South Africa in the spring but these weekends may become available depending on the severity of the Covid situation and associated travel restrictions.
The 1872 Cup matches, traditionally played back-to-back over the festive period, are the highlight of the domestic season but this week’s games were due to go ahead without fans due to the Scottish Government’s decision to limit live outdoor events to 500 spectators.
The new restrictions will be in place for “a period up to three weeks” which creates the possibility of rescheduled matches being played in front of supporters, a potential silver lining.
Edinburgh have also been hit by a number of Covid cases and head coach Mike Blair was due to miss the first leg because he was isolating as a close contact. Blair is expected to return to training this week.
Glasgow looked in good shape on Boxing Day when they were able to name a team containing 14 of the 15 players who started their previous match, against Exeter Chiefs.
However, several players returned positive lateral flow tests on the morning of the Edinburgh game and a round of PCR testing confirmed the results.
The players who tested positive will have to isolate for ten days.
A statement from the URC said: “The game was due to take place on Monday, December 27 in Glasgow, however, a number of positive Covid-19 cases within the Glasgow Warriors squad have been reported by the club.
“The URC Medical Advisory Group has liaised with Glasgow Warriors and Public Health Scotland and deemed that the fixture cannot go ahead as scheduled.
“The United Rugby Championship will now consider available dates to reschedule the game.”
Glasgow Warriors said they would “provide more information to season ticket members and ticket purchasers in due course”, adding, “we will continue dialogue with the league and Public Health Scotland in the coming weeks. The club thanks its supporters for their continued patience at this time.”