It was announced last year that youth charity Catch 22 and social care charity Turning Point had teamed up with support service company Serco in a £600m contract to run two prisons.
Navca and the Howard League have been vocal opponents of such developments, claiming that involvement in punishment is contrary to the ethos of charity.
A spokeswoman for the commission said the regulator was interested in the issue.
"We will be carrying out some fact-finding in order to gain greater understanding," she said. "Many charities can and do further a broad range of charitable aims by working in and around prisons - such as by advising on prison reform, offering welfare services to prisoners and their families, working around substance misuse or running education and rehabilitation programmes."
Frances Crook, director of the Howard League, told Third Sector that the involvement of charities in prisons would not only lower charities in public estimation but could also make prisons appear less forbidding.
"Prisons are places of punishment," she said. "They are dreadful things in a Dickensian sense and should be awesome and different."
She said she wanted the commission to forbid charities from being involved in running prisons. "This is a big issue of national importance," she said. "It is about the principle. It could be that other charities are looking at running prisons, but if you want to run a business, run a business, not a charity."