A petition for a "soft opt-out" procedure, where consent to donation is presumed unless the deceased indicated otherwise in life, has 18,000 signatures.
Labour has urged the Scottish Government to bring forward legislation similar to proposals going through the Welsh Assembly.
If ministers fail to act, Labour will take the initiative through a Members Bill, health spokeswoman Jackie Baillie said.
MSPs also faced calls to make more information available about risks associated with organ donation.
The family of a woman who died from skin cancer following a lung transplant told Holyrood's Public Petitions Committee they were unaware of the cancer risks associated with a common anti-rejection drug.
Ms Baillie said: "For me the clinchers was opt-out systems in other countries show a 30% increase on donation rates, and that frankly means saving lives."
The soft opt-out petition, lodged by The Herald's sister paper, the Evening Times, was continued for further evidence.