The scientists found 37 different viruses they believe have the potential to spread across the globe.
All of them have shown the ability to spread between people, but have not so far caused a major epidemic.
The Mers coronavirus, relatives of the Ebola virus, and several mosquito-borne viruses are singled out by the study.
Researchers said these viruses had all caused disease outbreaks in the past and were the cause of "greatest concern".
The method the team used to identify the most dangerous viruses has already predicted the threat of both the Ebola and Zika viruses before they emerged to cause major epidemics.
These infections are all zoonotic - meaning that they mostly affect animals at present. However, scientists fear they could pose a major threat to human health if they become able to spread more easily between people.
The research says surveillance of these viruses should be stepped up to avoid major public health crises, such as that seen in west Africa during the recent Ebola outbreak.
Prof Mark Woolhouse, one of the lead authors of the study, said: "Monitoring these infections should be prioritised because relatively minor changes in their ecology could lead to major changes in the threat they pose to public health."
A pandemic is generally defined as an epidemic that occurs over a very wide area, crossing international boundaries, and affecting a large number of people.
One of the most devastating pandemics was the Black Death, which killed an estimated 50 million people in 1350. Other more recent pandemics include HIV and the H1N1 flu virus outbreaks in 1918 and 2009.
The Edinburgh study was published in the journal Emerging Infectious Diseases.

Birmingham and Solihull Mental Health NHS Foundation Trust runs services from over 50 sites.
Inspectors criticised the safety of medicines management and said there was a high use of prone - or face-down - restraint.
Bosses at the trust said improvements were being made.
See more stories from across Birmingham and the Black Country here
The inspection by the Care Quality Commission (CQC) rated the trust as "requires improvement" overall. It was previously rated as "good", in 2014.
Inspectors said:
However, inspectors highlighted several areas of good practice and said staff were caring, compassionate and kind and treated patents with dignity and respect.
The CQC added staffing levels were generally safe and sufficient to provide good care.
The trust said it has already made improvements, including appointing a medication safety officer.
Chief executive John Short, said: "We recognise that the CQC has identified some areas where we did not meet the high standards we set ourselves.
"Since the inspection, work has already been completed in a number of key areas and is under way to address the other concerns raised."
The inspection took place between 27 and 31 March.
Experts at the University of Edinburgh have pinpointed more than 30 infections that are likely candidates for the next major pandemic.
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A mental health trust that provides services for 1.2m people in the West Midlands has been told by a health watchdog it "requires improvement".