The former Premier League boss also believes the City squad needs two to three more additions to be title challengers.
The Chilean came under increased pressure last season as his side failed to defend their league title, while they were also knocked out early in both domestic cup competitions.
Their relatively poor display in the Champions League didn't help, a competition those higher up at the Etihad Stadium are very keen to get their hands on at some point.
And Redknapp believes Pellegrini will need to either win the title or enjoy European success if he is to keep his job as Manchester City manager.
"To keep his job Pellegrini has to win something this year," he wrote on Kicca.
"I’m not talking about the Carling Cup, even the FA Cup for a club of that status isn’t enough. They’ll want to win one of the two big competitions, the Premier League or Champions League."
Two or three new signings needed
The former Tottenham manager, however, doesn’t believe the current City squad is good enough to win the Premier League title.
Pellegrini has been relatively quiet in the transfer market so far this summer, bringing in England duo Raheem Sterling and Fabian Delph, while Fulham youngster Patrick Roberts has also joined the squad.
While Sterling arrived for a big-money fee, Manchester City have arguably paid for his potential more than anything, and Redknapp believes they need another couple of big signings to be real title contenders.
"At the moment I look at them and I don’t really see them winning the title unless they go out and make a couple of big signings," he added.
"Raheem Sterling is a good young player with great ability and could be a big player for them this year. But I still think they’re short in one or two areas, I think they need to strengthen a bit more."