The Prison Reform Trust said it had received some calls from prisoners about their safety.
Chief executive Pia Sinha said this was “not unusual”, but “some of the calls have been worrying and concerning”.
The trust raised safeguarding concerns with the prison, which had been “responsive”.
Sinha said some of the challenges Millsike was facing had been seen in other new prisons.
“We do need to cut them a bit of slack because it takes time,” she added.
Mitie said safety was the “highest priority” and it maintained “a safe, secure and supportive environment for prisoners, colleagues and visitors”.
PeoplePlus, which delivers rehabilitation programmes, said: “Education and industry services have developed in step with the prison’s population.”
The MoJ said Millsike was “a vital part of our plans to build 14,000 new prison places”.
“A robust procurement process” had been conducted before the contract was awarded and it would “act when standards fall short”.
The has been told an annual report by the IMB, due to be published within the next month, will address safety concerns.