SNP MP Tommy Sheppard has reportedly written to the chief constable of Police Scotland asking to be kept informed of any investigation.
Campaigners have long accused the DWP's fit-to-work tests of causing deaths and have pointed to a number of suicides that took place in light of sanctions.
The Department for Work and Pensions said it would not comment on a police matter.
“It is important we make sure that people are receiving the right support, and they are not simply written off to a life on benefits," a spokesperson said.
"The Work Capability Assessment has been improved dramatically since 2008 following a number of reviews, including five independent ones.”
/ Will Frampton /