Several family genealogies claim, without documentation, that Asahel is the son of Rufus Stanton. That line goes back to a Thomas Stanton an early prominent settler in Connecticut. However 2010 DNA tests by a distant cousin (John Stanton) show that he (and we) are not connected to the Thomas Stanton line. Further evidence casts doubt on Rufus being Asahel's father. Asahel is not included in Rufus Stanton's household in the 1790 federal census when Asahel would have been 12 years old.
An alternate theory says "Two brothers named Stanton came to America from Ireland. One of them married a German girl and Asahel was one of their several children." This has been generally dismissed, but MAY be correct.
I've deleted the Asahel/Rufus connection in my personal genealogy.