Swinney said it was a valuable meeting and he agreed to meet again after further dialogue with the Scottish Government and JHI management.
Management at the Institute announced a consultation on a redundancy and restructuring programme that is expected to lead to a significant job cuts in September.
The James Hutton Institute is a well-respected and globally recognised research organisation delivering fundamental and applied science to drive the sustainable use of land and natural resources.
The job losses will affect staff at the Institute’s Aberdeen, Dundee and Edinburgh sites.
Prospect negotiator Ian Perth said: “It’s always heartening to engage with a senior politician who has a serious grasp of detail and genuine concern for our members.
“Prospect’s priority is to seek to protect our members from compulsory redundancies and to ensure that all those who remain in the Institute will remain on a sustainable basis.
“We welcome the employer’s decision to initially open a voluntary register and to offer enhanced redundancy terms for those who are attracted by voluntary exit.”
Perth also pointed out that in 2016, staff made a £1.7m annual contribution to the organisation’s finances by agreeing to significant pension changes and giving up their membership of the civil service defined benefit scheme.
The institute, which is primarily funded by the Scottish Government to carry out research on its behalf, has suffered significant budget cuts in recent years.