Smith was headed to Seattle on Thursday night following a visit with the Los Angeles Chargers, the New York Daily News reported, citing a source.
The Seahawks have a major need behind starter Russell Wilson following Tuesday's release of Trevone Boykin, who was arrested on domestic assault charges in Texas.
Smith has spent his entire career with the New York teams, playing his first four seasons with the Jets and last year with the Giants.
A second-round draft pick of the Jets in 2013, Smith started all 16 games as a rookie, throwing for 3,046 yards with 12 touchdowns and 21 interceptions.
Smith had 13 scoring passes against 13 picks in 13 starts in 2014, but he has made only two starts over the past three seasons, including one last season with the Giants.
In his five seasons, Smith is 12-19 as a starter and has thrown for 29 touchdowns and 36 interceptions while competing 57.9 percent of his passes.