Evans and Paterson both left the field early during Saturday's Six Nations defeat to Wales following respective neck and kidney damage.
Edinburgh full back Paterson, who became the first Scot to reach 100 caps for his country, is expected to be out of action for 10 weeks, although he could be released from hospital as early as Tuesday of this week.
Glasgow wing Evans underwent surgery to repair a damaged vertebrae and that operation has been deemed a success.
The 24-year-old has received visitors and is hoping to return to Scotland at the end of this week.
"The surgery that Thom required - having sustained damage to his neck - went well," explained Scotland doctor James Robson.
"Chris will also remain in hospital today after the damage he sustained to his right kidney. Fortunately he does not require surgery at present.
"In effect he has a massive bruise in the middle of his kidney but that will gradually repair itself and the medical team are also pleased with his progress.
"Thom had sustained slippage to one of the vertebrae and that required to be eased back into place. The surgeons are pleased with his progress, Thom has movement in his arms and legs and his family have been to visit him today, as have a number of his Glasgow Warriors team-mates.
"He will remain at the hospital today and if he continues to progress he will be allowed to travel back to Scotland into the care of the Glasgow Warriors medical team later this week."
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