Sandwell and West Birmingham NHS Trust's chief executive, Richard Beeken, said the trust's hospitals had 60 coronavirus inpatients, eight of whom were "critically unwell".
In a letter to staff, he said it was vital people had the vaccination to boost their protection, the BBC reported.
But out of 600 people aged under 40 who were called to encourage them to get jabbed, fewer than 10 agreed to make appointments.
Mr Beeken wrote that intensive care units were full and urgent cancer surgery was "under threat" due to Covid pressures faced by the trust.
The internal communication, seen by the BBC, suggested part of the issue was down to a "deprived and diverse population who have a very high vaccine hesitancy", according to the report.
The trust runs City Hospital in Birmingham and Sandwell General Hospital, in West Bromwich.
Staffordshire's hospital trust is also seeing a rise in Covid patients.
A spokesman for UHNM said: “University Hospitals of North Midlands has started to see a slow increase in the number of patients with Covid-19 being cared for in its hospitals as the rise in Covid-19 cases in the community across Staffordshire and Stoke-on-Trent continues.
“This is a situation which is constantly under review and we take all the appropriate steps to ensure our hospitals are safe for all our patients.
“It is extremely important that everyone plays their part in helping to limit the spread of infection and follow the Government guidance and remember Hands, Face, Space and Fresh Air and take up the opportunity to be vaccinated.”