The 27-year-old was 12 months into a three-year deal at the Dutch club but penned terms with Leeds on Friday morning after flying in from Krakow and undergoing a medical on Thursday.
Klich was a regular in Twente’s starting line-up last season, helping them finish seventh in the Eredivisie table and chipping in with eight goals and six assists.
The central midfielder is thought to have cost a seven-figure fee in the region of £1.5m and his arrival follows the permanent signing of winger Hadi Sacko whose loan move from Sporting Lisbon was turned into a permanent deal earlier this month.
Klich has been capped 10 times by the Polish national side and came on as a 45th-minute substitute in the 2-0 loss to England at Wembley in a World Cup Qualifier in October, 2013.
The new Whites recruit began his professional career at Polish club KS Cracovia before moving on in 2011 to German side Vfl Wolfsburg, who sold the midfielder to Dutch outfit PEC Zwolle in 2013.
After moving back to Wolfsburg in 2014, Klich signed for fellow German club FC Kaiserslautern in January 2015 before making the switch to Twente last summer.