Manchester City forked out £100m to snare Grealish away from Aston Villa earlier this month and he made his debut in the Community Shield defeat to Leicester City.
Kane has also been the other top target for Manchester City, but they have faced tougher resistance from Tottenham who are adamant that they will not be selling the striker.
The England captain has made it clear that he wants to leave but Tottenham’s position has remained unchanged with eleven days left in the window.
Carragher stressed that it is unfathomable that Manchester City prioritised signing Grealish before forking out the fee needed to sign Kane.
He stressed that it is clear from their defeat in the Champions League final last season and their opening day loss at Tottenham that Manchester City need a striker in their squad.
The former defender pointed out that Pep Guardiola has other players of Grealish’s quality and he is not going to make the difference in Manchester City winning or not winning the league this season.
Carragher wrote in his column for the Daily Telegraph: “City obviously had the funds to buy Kane for £150m at the start of the summer but decided to match the £100m clause for Jack Grealish.
“At the moment, it looks like they got their priorities wrong.
“When City lost the Champions League final to Chelsea, there was a broad consensus on what they needed to sign before the new season – a striker to replace Sergio Aguero and a recognised, consistent left-back.
“Nobody watching Guardiola’s side regain the Premier League title and go close to the European Cup was analysing their performances and saying City needed another attacking midfielder.
“Grealish is a fine player. He will do well at City. I understand why he wanted to move.
“What I cannot get my head around is why City did not attend to their more pressing needs first. City already have so many attacking midfielders of Grealish’s profile, Phil Foden, Ilkay Gundogan, Kevin De Bruyne and, if he stays, Bernardo Silva.
“Signing Grealish will not be the difference between City defending their title or finally winning the Champions League this season.
“Signing Kane, or a world-class striker capable of scoring between 20 or 30 goals this season, might be.
“That was clear in last weekend’s defeat to Spurs.”
Manchester City are still hopeful that they will be able to snare Kane away from Tottenham before the end of the window.
