Barry completed a £12 million move to Eastlands this week to become Mark Hughes' first summer signing but Redknapp revealed that Spurs weren't the only other interested party.
Redknapp has said that Liverpool, who were in pursuit of the England international last season, were also denied by City's financial muscle.
"This week, I tried to sign Gareth Barry," Redknapp told The Sun. "I felt he would be a good addition for Spurs and I knew we had to make a decent offer for a top England player.
"Liverpool were also prepared to make a big offer. But we have both been blown out of the water by Manchester City.
"They offered a far bigger transfer fee and are giving the lad much bigger wages. We could not get near."
Redknapp fears City's financial clout could help them win the Premier League title before long.
He added: "It wouldn't surprise me if they won the title quite soon as they seem to have more money than everybody else."
Redknapp also hit out at the money in football, saying: "While I benefit from a good wage, I'm embarrassed about it, to be honest. A lot of people in football are embarrassed."