Verweij, who lives and trains in Heerenveen , Netherlands, first attracted attention with a fifth place over 10,000 meters at the Dutch championships in December 2007. With this result he qualified as a substitute for the all-around European championships in 2008. At the speed skating junior world championships in 2007 in Changchun , China , he won the silver medal behind Jan Blokhuijsen . The following year he secured his first junior title in Zakopane , which he was able to defend at the 2010 Junior World Championships in Moscow . At the Dutch championships in 2009 he finished fourth and qualified directly for the all- around European championship in 2009 in his hometown Heerenveen. After falling over 500 meters in the race, however, he was only 12th and last in the final. After his successes in the winter of 2008/09, he received the Egbert van 't Oever Aanmoedigingsprijs for the second time after 2006 for the most successful young Dutch speed skater. For the 2009/10 season, Verweij started in the speed skating World Cup for the first time . In the following year he switched to the Hart team and was trained there by Jan van Veen .
At the 2012 World Championships in Heerenveen , Verweij won the gold medal in the team pursuit together with Sven Kramer and Jan Blokhuijsen. In 2013, the team was able to successfully defend its title with the same line-up. At the 2014 Winter Olympics in Sochi , he won the silver medal behind Poland's Zbigniew Bródka over 1,500 meters, just 0.003 seconds behind. In the team pursuit he was Olympic champion with Blokhuijsen and Kramer. A week after the Olympics, Verweij won the Dutch all-round title thanks to victories in the 500, 5000, 1500 and a second place over 10,000 meters. For his success at the Olympic Games he was awarded the Order of Orange-Nassau as a knight.
After Verweij had been practicing his sport self-financed without sponsors since 2015, he announced in 2018 that he would take an international break in winter 2018/19 due to lack of financial support.