Morris will be switching to McLaren when he is released from his contract at Sauber, and will work under its technical director Tim Goss.
A McLaren spokesperson said: "Matt Morris will join McLaren Racing as engineering director once his current contractual situation with Sauber is resolved.
"Matt will work for our technical director, Tim Goss, to ensure our engineering standards and technical decision-making capabilities are of the highest quality.
"During a period of intense technical challenge within Formula 1, Matt's appointment is the first step in an ongoing program to strengthen the engineering department as our organization prepares for the arrival of Honda in 2015."
06/28/13 (GMM) Sauber's chief designer Matt Morris is leaving the Swiss team, according to international media reports.
The German-language Speed Week, Blick and Auto Motor und Sport report that Briton Morris, 39, will be replaced by a deputy at the Hinwil based team, Eric Gandelin, a Frenchman who has been head of concept design.
Speed Week reports that Morris tendered his resignation, apparently after receiving an offer from another team.
Gandelin will reportedly start work on 1 July, becoming the third Sauber designer in as many years, in the wake of Morris and James Key before him.
Auto Motor und Sport said it has heard Morris, who has worked previously for Cosworth and Williams, is heading to McLaren.
Meanwhile, Sauber team boss Monisha Kaltenborn revealed she "expects" the team to stay with Ferrari power in 2014 as F1 moves from V8 to turbo V6 engines.