NFL Network's Ian Rapoport reported the pick involved in the trade is a sixth-rounder.
Since Rodgers missed significant time with a broken collarbone last season, Hundley started nine games for the Packers in 2017.
Hundley went 3-6 and completed 60.8 percent of his passes for 1,836 yards, nine touchdowns and 12 interceptions. He also rushed for 270 yards and two scores.
Green Bay originally selected the 25-year-old in the fifth round of the 2015 NFL draft out of UCLA.
Hundley will join a quarterback room in Seattle that already includes starter Russell Wilson, veteran backup Austin Davis and rookie seventh-round pick Alex McGough.
In three preseason appearances, Hundley has thrown for 263 yards, one touchdown and one interception. He's also rushed for 17 yards and one touchdown.
Since Davis has struggled in limited playing time (56 yards, no touchdowns and one interception) and McGough has impressed (285 yards, two touchdowns and one interception), Hundley's acquisition could spell the end of Davis' tenure in Seattle.
While Kizer has struggled with his accuracy during the preseason (50.9 percent), his numbers have been solid overall with 403 yards, two touchdowns and no interceptions.
As a rookie second-round pick last season, Kizer started 15 games for the Browns, going 0-15.
He completed 53.6 percent of his passes for 2,894 yards, 11 touchdowns and a league-worst 22 interceptions. The Notre Dame product did rush for 419 yards and five scores, however.
Trading Hundley likely means the Packers will keep only one quarterback behind Rodgers, which will give them more roster flexibility elsewhere.
Meanwhile, the Seahawks now have a backup with extensive, recent starting experience who can step in should an injury befall Wilson.