Claver began his playing career with the youth teams of his school, Sagrado Corazón HH. Maristas de Valencia, in Spain. He made his professional debut with Valencia during the 2006–07 season. He played in Europe's second-tier level competition, the EuroCup, for the first time during the 2007–08 season, and won it in the 2009–10 season. He also played in the European top-tier level, EuroLeague, during the 2010–11 season, and averaged 10.9 points and 4.9 rebounds, over 14 games.
NBA
Claver was drafted 22nd overall by the Portland Trail Blazers in the 2009 NBA draft. On 12 July 2012, Claver signed a three-year deal with the Trail Blazers. He was assigned to the NBA D-League's Idaho Stampede on 7 December 2012, and recalled on 9 December 2012. The following day, he made his first NBA start, scoring 4 points in 25 minutes with 5 rebounds and 2 assists. Claver was reassigned to the Idaho Stampede on 14 December 2012, and recalled on 16 December 2012. He played 80 games for the Trail Blazers over the next three years. On 19 February 2015, Claver was traded, along with Will Barton, Thomas Robinson and a lottery-protected 2016 first-round pick, to the Denver Nuggets in exchange for Arron Afflalo and Alonzo Gee. Three days later, he was waived by the Nuggets before playing in a game for them.
Russia
On 1 March 2015, Claver signed with BC Khimki of Russia for the rest of the season. On 17 July 2015 Claver and Khimki parted ways.
Attempt to return to Spain
In August 2015, he had a deal with Baskonia, but Valencia using the right of first refusal, matched the offer, and kept his rights in Spain. Baskonia obviously was not eager to pay a buy out to Valencia, but the Spanish forward did not want to play in his old team. According to reports, he did not want to play in a team which in the next season was going to compete in the second-tier level EuroCup, which led to Claver signing with a team outside the Spanish borders.
Escape to Russia
On 2 September 2015, the Russian team Lokomotiv Kuban announced a one-year deal with Claver for the 2015–16 season.
Return to Spain
On 20 July 2016, he signed a three-year deal with FC Barcelona Lassa, after his new team reached an agreement with Valencia for acquiring his rights. On 2 July 2019, he extended his contract with FC Barcelona through 2022.
