Suppose given a bass melody with either thoroughbass figuring or Roman numeral analysis. Is there a way to quasi-instantly recognize whether this bass melody admits of a canonic (or more generally, imitative) elaboration which is consistent with the supplied harmonies? For simplicity's sake let's assume that the bass melody consists of equal note values and that we're only interested in canons and imitations by direct (i.e., non-contrary and non-retrograde) motion. This essentially reduces my question to a problem in (tonal) first-species counterpoint.
I hope I understand your question correctly ...
Practically every bass melody can be used and composed as canon. Many canons are built by a) bass line and 2 or 3 upper parts like this canon:
Finally the canon is just a linear row of the chained voices: Dona nobis pacem:
If you have a cadence I IV V I or the progression I vi ii V you’re already on the save side:
I'm not sure this is what you are looking for, but it might help get to the point about determining the potential for canon, or you might say understanding the rule for a canon.
This video is where I first learned about a "rule" for canon at the fifth...
I also found this article that gets into the same idea in greater detail...
The main idea seems to be setting out two voices dux and comes (leader and follower) and then make a table of intervals that are permitted given the interval of imitation. So, canon at the fifth allows...
...to produce consonances between the voices. This comes from a chart in the Collins article.
Those moves present certain imitation/harmony associations. For example the second one - dux descending by step - creates a
I | V6 harmony. This seems to get at part of your question. A root position chord moving by descending step in the bass to a chord of the sixth has potential for canon at the fifth. Even more generally, a part moving by ascending thirds and descending steps has that potential. Of course, those aren't all inclusive for the full table of intervals, but they are patterns which should be easy to find.
I'm not sure about connecting this kind of Renaissance canon to functional harmony. The fifth example looks like
I | I6. In 18th century harmony I would expect the lower voice
E2 to move up to
F2, but the interval table does not allow for an ascending step. I haven't studied this enough to really understand it and give advice, but it seems connected to your question. Maybe it can help you.