It's possible, but there are alternative explanations.
Frankly, I don't believe that they draw on their godly power at all until they go SSB; rather, their bodies have simply become stronger from exposure to it. Their power has been consistent, but overblown IMO.
In the case of his fight with Beerus, it's possible that the power just hadn't worn off completely yet. He definitely became weaker, though, and Beerus started using less of his power to beat him up. However, once pushed to his limits, even without training he began to tap into his SSG power again as he punched Beerus back into space (according to Beerus). What I make of this is that there was still latent godly ki within Goku, more than he has now, the only reason he wasn't able to turn into SSB then was lack of ability to harness it.