The four-year maintenance contract is being advertised in the OJEU by the West London Mental Health Trust.
The successful bidder will be on call to deliver reactive maintenance at the crumbling hospital which is home to Ripper Peter Sutcliffe and Stockwell Strangler Kenneth Erskine.
The original hospital was built in 1863 and plans have been put forward for a £288m rebuild.
Individual jobs under the maintenance programme are expected to be worth up to £50,000 with an annual spend of £300,000.
Firms have until June 18 to express an interest in the job with tender invitations due out in August.