The striker penned new terms last month, but not before being linked with major clubs across Europe.
When asked if he had considered quitting Liverpool, however, Brewster told Four Four Two: “Yes I did, because I wanted to be playing first-team football and if that meant going, I'd have gone.
“But I felt it was right to stay at Liverpool and prove that I'm good enough to play in the first team.
“I hope to make my debut and play quite a bit this season. I've been injured since January; that's been hard, but Alex Oxlade-Chamberlain has helped me as we've been doing rehab together, and the manager has integrated me with the first team even though I've been injured."