Human-rights groups told the Associated Press news agency that many detainees are at risk of dying and have accused police officers of torturing prisoners and ignoring court orders demanding prisoners receive private medical care.
Fears the long-anticipated coalition government deal will not work will be heightened further if Mugabe, leader of the ruling Zanu-PF party, does not comply with Tsvangirai's demand.
He finally agreed to form a coalition with Mugabwe last week after months of dispute over who would effectively control key ministries - such as security and finance.
Allies versus technocrats?
Tsvangirai devoted much of Tuesday's news conference to naming his 21 appointments to the joint cabinet that is expected to be sworn in on Friday.
Tendai Biti, founding member and general secretary of the MDC, was named as finance minister - a key appointment given Zimbabwe's economic collapse.
"We are aware of the shortcomings... [of the unity government] but we are also aware of the potential"
Morgan Tsvangirai, opposition leader
However, in appointing Biti, a staunch political ally and vehement critic of Mugabe, Tsvangirai risks reinforcing fears that both leaders will select political allies rather than technocrats and economists.
Giving Biti the finance brief has also raised fears of confrontations between him and Gideon Gono, governor of the Central Bank and Mugabe ally.
Susan Booysen, a political analyst at the University of Witwaterstrand in South Africa, said working with Gono would be Biti's "major challenge".
"It is really going to take everything for him to show his mettle and to have a very good strategy to work out these obstacles," she said.
"I hope that at the end of the day, the sceptics will understand why we have done this in this political environment."
Of late, Zimbabwe has been devastated by hyper-inflation, food shortages and a cholera epidemic that has killed more that 3,300 people since August last year.