In the OED it is listed as sense 6 of allow.
Allow has had a diverse history in English since it emerged from the French alouer in the post-Norman conquest period. Its modern widely used senses involve "to take into account" or "to permit", or sometimes "to acknowledge" - but senses now less used involved things like "to receive with favour".
But sense 6 seems to me what you are looking for. It undoubtedly has its origins in southern England - the Sussex dialect seems its likely provenance - but today is mostly confined to the US "South and Midland" (I assume that covers Tennessee).
I have to admit that though I have lived nearly all my life somewhere in southern England I had never before noticed it. Next time I'm in a Sussex village I'll listen out for it!
- trans. Chiefly U.S. regional (chiefly south. and Midland) and Eng. regional (south.). With clause as object. To declare, claim, or state
(that something is the case); to come to a specified conclusion.
1738 S.-Carolina Gaz. 26 Jan. 2/1 A large Bell.., which by good
Judges is allowed to have the loudest and most melodious Sound of any
Bell in the Country.
1794 N. Parry Jrnl. in Kentucky Hist. Soc. Reg. (1936) 34 388 We
did not start until 10: having another Wilderness of ridges,
mountains, hills..allowed to be somewhat dangerous.
1825 J. Neal Brother Jonathan I. ii. 28 Her large eyes would
sparkle—so the men ‘allowed’—like the mischief.
1871 G. A. Townsend Mormon Trials 13 Well, in the first place, he
allowed he was doing his religious duties, and he allowed that he had
got to live with some one else.
1875 W. D. Parish Dict. Sussex Dial. 13 Master Nappet, he allowed
that it was almost too bad.
1913 F. W. Becker Home Rule of Eliza xv. 249 As I never cared fer
onnecessary trouble, I always allowed that the fust one to git to me
would jest have to fight them other two and settle it that way.
1956 Life 9 July 48 At the conference he allowed that he was
available as a candidate for President or Vice President if anybody
2009 T. Cheche Exit 10 vii. 126 She allowed as how we were welcome
to avail ourselves of their basement rec room.