Today, they can sense that it will rain soon.
Yesterday, they could sense that it would rain soon.
Today, they can sense that it is about to rain.
Yesterday, they could sense that it was about to rain.
Today, he can sense that his mother is about to bombard him with new information.
Yesterday, he could sense that his mother was about to bombard him with new information.
Today, he can sense that his mother will bombard him with new information.
Yesterday, he could sense that his mother would bombard him with new information.
The jarring semantic issue with will|would bombard him with new information (they are OK grammatically) is that "bombard" and "new information" are a little too specific to be "sensed" or "felt".