Police are investigating four deaths at Pentland Hill nursing home in Edinburgh and standards watchdog the Care Inspectorate is also looking at four separate complaints about quality of care which are unrelated to the police inquiry.
Last month, inspectors said the home must show immediate evidence of significant improvement or face closure.
The Care Inspectorate now says it is improving but more still needs to be done. A formal improvement notice issued in August will remain in place until a further inspection in the new year.
A Care Inspectorate spokesman said: "The health, safety and wellbeing of residents is the absolute priority which is why we are taking enforcement action against Pentland Hill, and working closely with the council and NHS to safeguard every resident.
"This home is not yet operating at the standards we require, but we have seen some evidence of some improvement. This is in the early stages and needs to be sustained."
Pentland Hill is run by Bupa.
Vivienne Birch, director of partnerships at the private healthcare group, said: "We're pleased the Care Inspectorate has seen improvements within the home. We have introduced a new leadership team, recruited more staff, invested in new equipment and made significant improvements to medication management and nutrition."