No citation but in times of inflation the reasoning makes sense to me: a slave would not earn a wage, but the owner would have to provide a roof/bed/food + pay for whatever transportation was needed to/from work + pay for healthcare in case the return on investment would be worth it (probably would?).
An (immigrant) worker gets none of that and might barely be able to get by even without counting the healthcare (in the US).
Sounds to me like a slave might indeed be cheaper in some/many situations than a minimum wage worker. I'm not convinced either way.
A migrant is more expensive when you have work to do. However a migrant is free when you have no work - they go elsewhere. A slave you need to feed year round, even when it is raining and thus you cannot work.
A slave also needs more management. Migrants and free workers will get themselves to the job and in general work. A slave has no motivation to work harder so you need some form of "slave driver" to keep them working. If you try to move your slaves around like migrants move, then you need a manager to go with the slaves to keep them working - migrants manage themselves.
A slave is cheaper if you have a lot of repetitive, low-skill, year round work that must be done by hand. However most of that type of labor is easy for the industrial revolution to automate.
It's weird that you think that they had it better than migrant laborers.