I've seen accusations from pro-slavery advocates that the white lower classes in Britain would be better off if they were enslaved, and I've seen accusations from Northern anti-slavery advocates that the ultimate logic of Southern slavery would be to enslave the white lower classes, but I've never actually seen a proposal from a Southern slave-owner that white people should be forced either into indentured servitude or full-blown slavery. The two names that have been put forward are George Fitzhugh and Robert Rhett, though nobody ever provides a quote from the latter to support the argument.
The former's book Sociology for the South argued "in countries where there are no negroes, we can see no reason why the whites in all cases might not be allowed to sell their persons for short periods" but "we need never have white slaves in the South, because we have black ones". He actually argued for the elevation of the poor whites on racial grounds:
"Educate all Southern whites, employ them, not as cooks, lacqueys[sic], ploughmen, and menials, but as independent freemen should be employed, and let negroes be strictly tied down to such callings as are unbecoming white men, and peace would be established between blacks and whites. The whites would find themselves elevated by the existence of negroes amongst us. Like the Roman citizen, the Southern white man would become a noble and a privileged character, and he would then like negroes and slavery, because his high position would be due to them."
I think it may be a myth, though I'm keeping an open mind on the topic.