On 8 April, Agüero scored a solo goal to give City the win in the Manchester derby in the 78th minute, seven minutes after coming on as a substitute for Samir Nasri. Agüero then scored a header to help Manchester City defeat Chelsea on 14 April in the FA Cup semi-final and reach the final. In the 82nd-minute of the tie, Agüero made a two-footed tackle on David Luiz after the defender had previously appeared to tackle him. Despite the incident warranting a red card at the time, the FA announced Agüero would not be punished following confirmation that referee Chris Foy had at least seen part of the incident and acted on it during the match. Many people disagreed with this decision and felt that the FA had failed to act. On 17 April, Agüero revealed that he had apologised to David Luiz for his rash challenge on the defender. Agüero withdrew as a precaution at half-time on 17 April in Manchester City's 1–0 home victory over Wigan after feeling tightness in a hamstring, but he appeared as an unused substitute in City's 3–1 away loss at White Hart Lane against Tottenham on 21 April. He scored his next goal on 27 April, where he scored the opening goal, as Manchester City won 2–1 home against West Ham United. On 11 May, Agüero started for Manchester City in the 2013 FA Cup Final, where the team lost 1–0 to Wigan. Agüero scored his final goal of the season in a 2–0 away win against Reading on 14 May.
The Times reported that he was the main transfer target of Spanish giants Real Madrid, but Agüero responded by saying, "I'm very happy at City and I feel appreciated and loved here which means a lot to me." Manchester City chief executive Ferran Soriano also negated the idea of a transfer, insisting Agüero would stay at the City of Manchester Stadium. However, former Argentina international Diego Maradona backed him to join Real Madrid, believing Agüero's presence in the Madrid team could help get even more out of Cristiano Ronaldo. On 25 May 2013, Agüero signed a one-year extension to his contract, keeping him at Manchester City until 2017. Agüero stated his commitment to Manchester City on 28 May 2013, saying, "I'm committed to Manchester City, it's a great club."
Following a knee injury that ruled him out of all of the club's pre-season games, Agüero scored in City's 4–0 victory over Newcastle United in the opening game of the 2013–14 Premier League season. On 22 September 2013, he scored a brace in a 4–1 home win against Manchester United.
On 8 November 2013, after scoring five goals in four matches between 5 October and 2 November, Agüero was awarded the Premier League Player of the Month for the first time. During the same period, he had also scored both goals for City in a 2–1 away win against CSKA Moscow in the Champions League.
On 14 December 2013, Agüero was substituted in a 6–3 win against Arsenal due to a calf injury and missed eight matches. On 16 January 2014, he returned from injury in an FA Cup third-round replay against Blackburn Rovers, scoring after coming on as a substitute in the 5–0 win. In the next round of the competition, he scored his first hat-trick of the season in a 4–2 victory over Watford. On 29 January, he scored his 50th Premier League goal in a 5–1 victory at Tottenham but was substituted with a hamstring injury.