There's a post on reddit regarding this. Though it is just an assumption, TS has done great work on calculating all the chances. From what I can see, the chances do scale up.
According to the graphs and tables presented in that discussion we can track that that overall proc rate increases from ~30% at the game start to the max of ~45% proc chance in a game that lasted for more than 52 mins. That is linear proc chance we can see in the graph. We can still observe spikes up to 70% in the mid to late game and down to less than 20% in the early to mid game.
Here I will cite the author: Looking at the data points, there are a couple things we can assume:
- The chance of Kleptomancy activating scales up with game time.
- Getting sustain potions scales down with game time.
- Getting combat potions, Control Wards, and Farsight Wards scales up with game time.
- Getting Minion Dematerializers, Sight Wards, and Potion of Rouges appear to stay relatively the same over the course of the game.