Last month, Mr Swinney announced plans to reverse a commitment to provide core and development funding to the Scottish Men’s Shed Association, which has supported the establishment and development of men's sheds across Scotland since 2014.
The decision has prompted more than 3600 people to sign a petition calling for a rethink.
In parliament, a motion lodged by Alex Rowley MSP has already gained support from MSPs representing every political party.
There are more than 200 men's sheds in Scotland across all 32 local authorities which support more than 10,000 men aged over 18 to develop new hobbies and skills, while helping tackle social isolation.
Jason Schroeder SMSA chief executive, said: "The SMSA has grown the Scottish Men's Sheds movement from an initiatory idea into the biggest and most successful men's health movement in Scotland. This is now the time to invest so we continue to not only provide much-needed stability post-Covid to existing and developing groups, but we are there to support the new Shed groups which continue to emerge.”