The Eastlands keeper revealed Mancini is under huge pressure to deliver silverware to repay the £200million spent on players by City's Arab owners.
And more cash is expected to be splashed with England midfielder James Milner on the wanted list.
Given said: "We're at the point where the club has invested so much and the manager understands and knows that the pressure is on to get results and trophies.
"City have the backing to challenge the top teams in England and even in Europe in years to come.
"I don't want to say we are going into the league to finish fourth.
"We are going in to win it and if we don't, the next thing will be the top four.