The intent is that your base score, not a temporary score, has to meet a multiclassing prerequisite.
While that tweet is Rules As Written when specifically referencing using temporary ability scores for multiclassing, I believe that it can be applied to this question as well.
From that tweet, we can tell that the designers have intended a distinction between your characters base ability score, and your characters temporarily modified ability score.
While I don't see anything in RAW that mentions Ability Score Improvements applying to the base score instead of the modified score, it would seem from that tweet that the designers did not intend the modified score to be used for level up related things, such as multi-classing and then by extension: ability score improvements.
Additionally, if it were to work in the way suggested, where the ASI increases your Int to 20, then that would imply some very strange rules interactions elsewhere. For instance, if you were to take the headband off and put it on again, your Int would be 19, not 20. That would be a somewhat strange interaction.
Even more strange would be, if you were under the effects of a Potion of Giant Strength (Sets your strength to 21) when you leveled up, RAW would not let you use your ASI to increase your strength, as you cannot increase an ability past 20, and the Giant Strength potion sets your strength to 21.
Both of those would be very strange rules interactions, especially the second one. I'm fairly certain the designers would not expect Potion of Giant Strength to prevent you from leveling. Therefore, given the way that rule would play out, as well as given the Crawford tweet about multi-classing, I believe we can conclude that the ASI was intended to modify your base ability score, not your temporary ability score.