Wright spent his formative years at Langwarrin, before playing a season each with Mornington and Dandenong Thunder. He was identified as a promising talent, eventually earning selection for the Victorian state team in 2007. He honed his skills further with the Victorian Institute of Sport, before moving overseas in July 2009, signing a two-year scholarship with Preston North End after a protracted clearance saga.

In his first season with Preston North End, Wright played predominantly as a centre-half for the Reserves and Under 18s teams, but was eventually called up to the senior squad in April, sitting on the bench for the club’s fixture away to Coventry City. He commenced the 2010–11 season with his debut senior appearance in the first round of the League Cup, playing the full 90 minutes in the Lilywhites’ 5–0 win against Stockport County. On 13 December 2010, Wright was handed a two and a half-year professional contract.

Wright made his league debut for Preston on 5 March 2011 against Norwich City. He scored his first professional goal for Preston on 7 April 2012 against MK Dons.

Wright was voted Young player of the year of Preston North End for the season 2012/13. On 1 May 2013, he was handed a new two-year contract, with the option of staying at Preston North End for another year.

Wright won promotion via the Play-Offs with Preston in 2015, defeating Swindon Town 4–0 at Wembley on 24 May 2015.

On 6 January 2017, it was announced that Wright joined Bristol City for an undisclosed fee, signing a 2 ⁄2-year contract. On 7 January 2017, he made his debut in the FA Cup Third Round Tie against Fleetwood Town. He scored his first goal for Bristol City in a 1–1 draw with Norwich City on 7 March 2017. At the start of the 2017–18 season, Wright was named Bristol City captain.

Wright featured as Bristol City reached the semi finals of the 2017–18 EFL Cup with wins over Premier League opponents Watford FC, Stoke City, Crystal Palace F.C. and Manchester United. Wright played as City lost in the semi-final tie against Premier League leaders Manchester City. Wright was released by Bristol City at the end of his contract in July 2020.

Wright signed a six-month loan deal with League One club Sunderland on 21 January 2020. On 2 August 2020, Wright signed for Sunderland on a permanent basis, on a two-year deal. He scored his first goal for Sunderland in a 2-2 draw with Rochdale on 27 October 2020.