The 28-year-old Glasgow Warriors midfield ace suffered the injury in the 1872 Cup clash victory over arch-rivals Edinburgh at Firhill and was forced to sit out the return game at Murrayfield and the narrow Magners League defeat by league leaders Munster on Saturday.
And over the weekend both Warriors coach Sean Lineen and Morrison himself confessed the knee problem could keep him sidelined for up to two months.
"I like to think of myself as a quick healer. I've done this on the other knee before and I was back before they predicted," Morrison told BBC Scotland.
"I'm hoping it will be around six to eight weeks, which would give me a chance to play in some of the Six Nations. I've been playing well this season, so to have this is a bit of a setback, but I'll just do my best to get back on the pitch."
The Warriors end their Pool 6 campaign with a home clash with London Wasps this weekend and a trip to Magners League rivals Newport Gwent Dragons in Round 6. Beyond that, Morrison will almost certainly miss Scotland's RBS 6 Nations opener against reigning Grand Slam champions France at Stade de France on Saturday, 5 February.
How much of the tournament he will be fit to play in will depend on how quickly he can recover.