On 6 January 2013, Suárez handled the ball before scoring the decisive goal in Liverpool's 2–1 win over Mansfield Town from the Conference National in an FA Cup third round match. Liverpool's manager Brendan Rodgers defended his player by claiming "it's not his job to own up", while Mansfield manager Paul Cox said he felt "a little bit gutted" by the "instinctive" handball, but admitted he would have accepted a goal scored like that by one of his players.
At the end of the season, Suárez was one of six players named on the shortlist for PFA Players' Player of the Year. Suárez finished second in the final ballot behind Gareth Bale of Tottenham and was named in the PFA Team of the Year. He was second-top goalscorer in the Premier League for 2012–13 with 23 goals and Liverpool's top scorer in all competitions with 30 goals. On 28 May 2013, he was named as Liverpool's player of the season after receiving 64% of votes in a poll of the club's supporters.
On 21 April 2013, during a 2–2 draw with Chelsea in a Premier League match at Anfield, Suárez bit Branislav Ivanović; this was the second time Suárez had bitten an opponent. It was not noticed by the officials, and Suárez scored an equaliser in injury time. The bite prompted UK Prime Minister David Cameron to call on the FA to take a hard line with Suárez: the FA charged him with violent conduct and he was fined an undisclosed sum by his club. Contrary to claims from Suárez, Ivanović did not accept an apology. Suárez accepted the violent conduct charge but denied the FA's claim the standard punishment of three matches was clearly insufficient for his offence. A three-man independent panel appointed by the FA decided on a ten-game ban for Suárez, who did not appeal the ban; the panel criticised Suárez for not appreciating "the seriousness" of the incident when he argued against a long ban. The panel also wanted to send a "strong message that such deplorable behaviours do not have a place in football", while noting that "all players in the higher level of the game are seen as role models, have the duty to act professionally and responsibly, and set the highest example of good conduct to the rest of the game – especially to young players".
On 31 May 2013, Suárez said he would be seeking an exit from Liverpool in the summer, citing excessive media attention on his family as a reason for wanting to leave. On 6 August, after Liverpool had rejected a bid of £40,000,001 for the player from Arsenal, Suárez reiterated his wish to leave Liverpool and said Liverpool had previously promised to allow him a transfer if the club failed to qualify for the 2013–14 UEFA Champions League. The following day, Liverpool manager Brendan Rodgers said Liverpool had not broken any promises to Suárez and that the player had shown "total disrespect" for the club. After this incident, the British press reported that Suárez had been instructed to train away from the Liverpool first-team squad by Rodgers. On 8 August, Liverpool owner John W. Henry stated that Suárez would not be allowed to leave the club.
Suárez ended the 2014 World Cup qualification campaign as top scorer in the South American section with 11 goals in 14 matches. In November 2013, Uruguay defeated Jordan 5–0 on aggregate in an intercontinental play-off to qualify for the 2014 World Cup.
Suárez has been widely accused of diving. His manager, teammates and various analysts have commented that this reputation for simulation has caused referees not to award him legitimate penalties. In January 2013, Suárez admitted to diving against Stoke City in an October 2012 match. This led his manager Brendan Rodgers to comment his actions were "unacceptable" and that he would be dealt with "internally" by the club. During the 2018 World Cup game against Portugal, after collisions with Portuguese players, Suarez twice acted like he had a head injury despite Suarez's head not being touched.
Due to his diving, biting, stamping, and other antics, Suárez has been frequently labelled as a pest and a cheat. In December 2013, Spanish football website El Gol Digital ranked Suárez at fifth in its list of the world's dirtiest footballers.