'El Toro' or The Bull is a nickname he earned for his quickness and physical play while he was in the Spanish Under-21 team. Enrique made his breakthrough at Villareal before joining Newcastle for £6.3 million after just one season with the Yellow Submarines. Enrique was selected as Newcastle's official Player of the Season for 2009/10 and named in the Championship's PFA Team of the Season as the Magpies won promotion to the Premier League. Two days before the start of the 2011/12 Premier League season, Liverpool purchased the Spanish left-back. Although Enrique still had one year left on his Newcastle contract, a move away from Tyneside was always likely after he had criticised the club’s transfer policy earlier in the summer.

Enrique joined a Liverpool squad already possessing a number of players either youthful or experienced who could occupy the position he was most suited to play in. However, the left-back position in the team was one that had been rather difficult to fill permanently in recent years. Kenny Dalglish was confident enough in his latest acquisition to put him straight into the first team on the opening day of a new Premier League season. By the turn of the year, he had impressed so much with his defensive stability and attacking surges upfield that he had played in all nineteen of the Premier League matches up to that time. His form dipped alarmingly towards the end of the season, but he was still a regular. He picked up a winners' medal in the League Cup and also had an unexpected seven minutes in goal against his old club Newcastle United after Pepe Reina had been sent off at a time when Liverpool had used all their three substitutes. Enrique has proved a key player for Liverpool since Brendan Rodgers took over, either used as a left-back or left-winger.

Enrique played in thirty-five of Liverpool's fifty-four competitive matches in 2012/13. He also scored his first Premier League goals for Liverpool in the home matches against Wigan Athletic and Swansea City, and in so doing he doubled the total of Premier League goals he had scored in four seasons as a Newcastle player. Injuries restricted Enrique to just 8 first-team appearances in 2013/14 and according to his tweets was quite busy honing his skills playing FIFA, in which he reigned reportedly as club champion.

Enrique only started six matches for Liverpool in the 2014/15 season but came on as a substitute on three more occasions. He passed his 29th birthday during the season, which ended with him still having one year left on his Liverpool contract. Enrique explained to Marca in the summer of 2015: "The only teams that can deal with what is paid in England are Real Madrid and Barcelona, so I would love to stay here until my legs stop me. I had a very good option to return to Spain, but is complicated because the economy is very different from what exists here.”. Despite being asked to train alone and excluded from Liverpool's pre-season preparations, Enrique failed to find another club and at the end of the transfer window vowed to fight for his place at Anfield. However, he was frozen out of the first-team picture by Brendan Rodgers and then by Jürgen Klopp who took over in October. A defensive injury crisis led to him playing at centre back in the FA Cup third-round tie at Exeter, where he failed to impress and was arguably at fault for the home side's first goal, his lack of pace being exposed. He played again in the repay, came on as an 89th-minute substitute against West Ham in the 4th round and never appeared for the Reds again. Enrique was unsurprisingly released at the end of the season, having been told before the final league game at West Bromwich Albion that he didn't even need to turn up for training any more. Although disappointed he did leave with no hard feelings, posting on social media that Liverpool's fans were the best in the world before going on holiday to Ibiza with Suso.

Enrique's contract at Liverpool expired in the summer of 2016 and in September he joined second-tier Real Zaragoza in his native Spain. He scored once in 27 appearances in Segunda B but announced his retirement on 6th September 2017 citing persistent knee problems as the reason. Enrique was diagnosed with chordoma in May 2018, which is a rare type of tumour that usually occurs in the bones of the skull base and spine. He had life-saving brain surgery after the tumour was discovered behind the eye. After his recovery, Enrique made his debut for Liverpool legends' team when he featured with former Reds against Borussia Dortmund in Hong Kong in June 2019. He went into punditry, became active on social media where his love for Liverpool FC is clear and also engages in motivational speaking.