Sterling joined the Reds back in 2010 from Queens Park Rangers and was given his debut in the dying embers of Kenny Dalglish's reign back in 2012.
It was under Brendan Rodgers that the youngster truly thrived, as he made 24 appearances in the 2012-2013 campaign before going on to play 33 times last term.
Sterling emerged as a key member of the Reds' ultimately unsuccessful title push in the final few months of last season, eventually finishing with nine goals for Rodgers's team, and his eye-catching form was rewarded with a spot in the England World Cup squad.
Still just 19, the versatile Sterling has benefited from being introduced into a more central attacking role by Rodgers, and reports that the teenager was in line for a huge pay-rise at Anfield have circulated over the last few months.
Speaking to the Irish Examiner, Brady, who used to work as Arsenal's academy boss, explained that he dropped his pursuit of Sterling prior to his move to Anfield.
The Irishman said: "In recent times, I think I could have got Raheem Sterling if I'd really gone for him, but I didn't go through with that one. Players we don't keep often go on and have a good career in the game.
"It isn't an end -- OK, he's leaving Arsenal, but clubs are always ringing us, saying: 'Well, tell us about this fella, and give me a list of the lads who are leaving.' That isn't said to soften the blow for the lad, that's the truth. Within weeks, these lads have moved to another academy.
"I saw a kid recently, Lewis Grabban. Norwich have paid around £3 million to Bournemouth for him - I let that kid go when he was 14. Dwight Gayle, who is playing for Crystal Palace, we let him go too; he cost Palace a couple of million."
Sterling will have added responsibility placed on his shoulders this coming campaign, following the £75m sale of Luis Suarez to Barcelona.
Reds boss Rodgers has bolstered his squad to the tune of £90m this summer with the likes of Adam Lallana, Rickie Lambert, Lazar Markovic, Emre Can, Dejan Lovren and Divock Origi - who will spend this coming season on loan at former club Lille.