In text, does 'enumerate' imply a numbered list?

From Merriam Webster’s page on enumerate:

1:  to ascertain the number of: Count

2:  to specify one after another: List

Also from Merriam Webster’s page on enumerate

Despite its numer- root, you don't have to use numbers when enumerating.

Based on the information provided by the dictionary’s page on the word, I can only conclude that you can enumerate your reasons without using numbers or giving the impression that you’re relying on a numbered list.

I would urge you to take a look at the Merriam Webster’s page on the said word.