Something like what you're describing did happen in the North, though it's not quite the same. With the exception of Massachusetts and New Hampshire, the future free states got rid of slavery by means of gradual emancipation. This was itself not an easy political task, because enslavers tended to be well off and powerful. New York had to take two tries at it. New Jersey didn't get their emancipation law passed until 1804. The major differences between the North and South on this point are that most northern whites don't understand slavery as a civilization-defining institution because there aren't enough enslavers or black Americans around. That changes quickly once you cross the Mason-Dixon.

But that's getting a little ahead of myself. Gradual emancipation is like what you're suggesting in some senses. The basic frame is simple: the legislature passes a law that sets a date at which emancipation begins. I say begins because no one is freed on that day. Rather, the law provides that everyone born to an enslaved mother on or after that date becomes free at a specified date in the future, usually when they're into their twenties. (Specifics vary state to state.) Until that time, these children and young adults are owned by their mother's enslaver, who is entitled to their labor. They spend those few decades "earning" their freedom that way, which compensates the enslaver for the infringement to his or her rights to property in people. Once the time is up, they're free. Anyone born before that date is a slave for life. This, naturally, results in "free" states which have enslaved people in relatively large numbers for decades after. With the exception of New Jersey, all the free states come back a few decades down the road from the initial enactment and pass another that frees everyone that was born too soon, who are usually quite old by that point and probably have aged out of maximum profit-generation.

This all worked, at least for a certain definition of worked. A fair bit of cheating and gamesmanship went on, but slavery did dwindle in the free states as the nineteenth century went on. The system is a nonstarter in the South, though versions of it do get proposed from time to time. White Americans in both sections share a powerful hatred of black people, to the point where western states try to ban the presence of blacks entirely. The thing is that white Yankees can imagine a lily white continent all to themselves because black Americans are a distant fact -someone else's problem- or exist in small numbers and largely concentrated in cities. All of this is impossible in the South, where even in relatively less enslaved states you can often find counties that are more than half black. From a white POV, any blacks are undesirable -intensely so- but white Yankees already live in such white spaces that they don't seem like an existential threat. White southerners do not live in that setting. Often locally outnumbered and required to imagine a far larger black population sharing space with them, even the most slavery-adverse among them -who usually aren't very much so, but sometimes talk a good game- they believe that any plan of emancipation needs to include removal of the freedpeople.

This hits up against practical difficulties, even if we had a white South that might have considered the option seriously. Transporting all the freedpeople to Liberia, Central America, or somewhere far, far away in the American West would cost a lot of money. Who is going to pay for that? Efforts by people in free states to make it a national project provoke fierce opposition even when confined to no more than resolutions passed in state legislatures suggesting the course. A federal government that had the power to do such a thing would also have the power to force emancipation on a reluctant South, or otherwise undermine slavery so as to make it inevitable, so white Southerners tend to oppose federal power on those (and usually only those) grounds. So it's got to be a local project, done by the people least inclined to do it and with their own treasuries.

They deem this impractical pretty much from the start, at least as a solution for the whole slavery question. I don't think anyone tried to run the numbers in detail until the proposal gets discussion during a Virginia constitutional convention. The result was the first canonical work of proslavery theory. Thomas Dew figured that the state could, maybe, afford to buy and transport a few thousand slaves a year. But enslaved mothers would keep having enslaved babies in that time and the state would never catch up with that rate of growth. Virginia could not produce that kind of cash, period. So even if you wanted to try, it's just not possible and best not risk it because talk of freeing enslaved people, white Southerners believed, was a great way to get yourself a slave revolt.

Simply buying all the nation's slaves with cash was beyond the means of the federal and state governments, even could the enslavers be convinced to sell. They would not because they're horrific racists, because they are making money on this, and also because they need the labor. The South did not have a large pool of laboring whites who would take the place of enslaved blacks and had little hope of enticing immigrants to come do the same work for less than it cost them to run slave labor camps. They could get some, but nowhere near enough to transition. So asking them to give up slavery is asking them to surrender the money fountain twice over: they have to give up their profit engine in both the immediate and foreseeable future.

Realistically? No, the South is not open to this form of emancipation or to any other. Enthusiasm for the project at all probably peaked immediately after the Revolution and a spent force by about 1800. There is a real boom in interest which we should not dismiss, but we do need to put hard qualifiers on it: emancipation is an interest of a minority of whites in and around the Chesapeake. From North Carolina on south, the overriding interest is in getting more slavery going.

All that said, there are small-scale, informal arrangements to this effect negotiated from time to time. An enslaver might cut a deal with an enslaved person that for a specified price, he or she can buy themselves (or a child, spouse, sibling, parent, etc) possibly provided "good behavior" and with an additional term of years' service. None of these arrangements is better than the enslaver's goodwill because the enslaved have no right to own property -so the enslaver can just take their cash at any time for any reason- and because the enslaved person can't take them to court or anything if they change their mind. That's best case. Enslavers are also vulnerable to economics beyond their control, which might put even the most frugal into debt. They may have to sell people to make their creditors whole, even if they otherwise intended to follow through. They might also die and have their human property inherited by people who refuse to honor the arrangement. Though enslavers do try to curb the practice, it was probably never within miles of common enough to pose a real threat to the system as a whole.