The 34-year-old goalkeeper penned a short-term deal with the Dons yesterday after being released by Sheffield Wednesday and is keen to get some game-time under his belt.
However, he is well aware he will have to work hard for any opportunities that come his way.
Weaver told RedTV: "I want to play but I'm sure Jamie will have something to say about that.
"He's been the number one here for quite some time and he's obviously a very able keeper. I think he played just about every game last year.
"I'm under no illusions. I don't feel as though I'm going to walk straight into the team.
"All I can do is my best in training. If and when I get an opportunity, it's up to me to take it."
Former Manchester City and Dundee United keeper Weaver is hoping for a better reception from Dons fans this time than he experienced the last time he was in Scotland.
He said: "My last game for Dundee United was against Aberdeen and I remember quite vividly I got pelted by snowballs.
"It was all a good bit of banter and I had a laugh about it then.
"I really enjoyed my time when I was here (in Scotland) the last time and it made the decision easier to come up again.
"Hopefully I can enjoy it and be part of a successful Aberdeen team."
On the prospect of potentially remaining at Pittodrie longer than his initial short-term deal, he said: "It's a six month deal and I think that suits everybody.
"I'm not really looking beyond that. All I want to do is knuckle down and do my best and hopefully play some games. The main thing is the team getting the right results."