"I think the criticism will be saved to the end of the season if results in big away games don't improve and it costs them the title", Carragher said on Sky Sports' Monday Night Football.
And the manager was adamant that there was no quality gap between his team and Manchester United, who are two points behind City in second place.
Mourinho's post match comments poking fun at Klopp and Liverpool's lack of urgency to win the match were created to shift attention away from a disappointing performance from his own side, as they failed to test an opposition defence that would have cracked if United had made the most of their first half openings and delivered better set-piece delivery when they had the opportunity.
"Liverpool were the better side by far".
The current side could not add another win over their bitter rivals at the weekend, with Mourinho's men holding out for a 0-0 draw.
The Merseyside giants have been guilty of squandering numerous goalscoring chances this season, while the defence has also let in soft goals. We thought they would have shown more ambition in the game. "If we were to play this way... we could not do it at Liverpool".
"Klopp came out after the game and said he would never get away with playing such negative tactics at Liverpool, but did anyone expect anything different from Mourinho", he asked. But Mourinho got the result he wanted. With 10 minutes to go, United looked there for the taking but Klopp took off Mo Salah and Philippe Coutinho - Liverpool's two most risky players on Saturday.
Liverpool, who are now without their injured star Sadio Mane, have scored 13 goals in the league in the present campaign, the fourth-highest in the division.
Given Mourinho is only 54, it was always going to be unlikely he would stay at Old Trafford for the remainder of his career in the manner Sir Alex Ferguson did, with long-term managerial reigns like those of the Scot and Arsene Wenger at Arsenal a dying breed.