Morrison was carried off the pitch just 13 minutes into the Greaves Sports 1872 Cup game against Edinburgh at Firhill with a knee injury.
It was first thought that the 28-year-old Scotland centre would be a long term casualty and a possible doubt for the entire RBS 6Nations. But Lineen is hopefull his powerful centre can make a quicker recovery than initially thought.
"I don't know how long he's going to be out but I understand at the moment it's weeks rather than months," said Lineen.
"He's a crucial player for us and for Scotland. I really feel for him. He's been playing really well for us, but he'll get the best treatment and he'll be back as soon as we can get him back."
Glasgow recorded a 30-18 victory in the first leg of their back-to-back Magners League encounters with their Scottish rivals and hope to follow that up with another triumph on Sunday.
Morrison is certain to miss that clash and faces a race to be fit for the RBS 6Nations opener with France on February 5.
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