In a letter, council leader Stuart Bray said he “warmly” welcomed improved NHS facilities in the town following decades of campaigning, but urged the board to “look at the plans again”.
A council motion urged the applicant to “look for ways to work with local heritage groups and others to seek to preserve as many heritage features as possible”.
Dr Luke Evans, Conservative MP for Hinckley and Bosworth, said: “In just a few short weeks, we have seen what a difference the £24m community diagnostic centre has made to people across our area, who can now get their tests, scans and checks right here in Hinckley.
“I know how important access to local healthcare is for people and families across our community, and look forward to seeing this exciting investment take shape now it has the green light.”
Toby Sanders, interim chief executive of the LLR ICB, said he was “absolutely delighted” the plans could now move forward.
“The development of the day case unit is the second phase of our development for Hinckley and will cater for a wide range of clinical specialities in modern facilities,” he said.
A spokesperson for the Department of Health and Social Care added: “As part of the 10-year health plan, we want to shift healthcare out of hospitals into the community, to ensure patients and their families receive personalised care in the right place, at the right time.
“We are pleased to be providing more than £7m for a day case unit in Hinckley which will greatly benefit patients and the community.”