A seventh-round pick by Dallas in the 2015 NFL draft, the 6-foot-4, 260-pound Swaim has 35 receptions for 336 yards and one touchdown in four seasons. He caught 26 balls for 242 yards and a score in 2018.
He is expected to serve primarily as a blocking tight end with the Jaguars, who opted not to keep Austin Seferian-Jenkins, James O'Shaughnessy and Blake Bell after last season.
Ben Koyack and Pharoah McKever had been the only tight ends on Jacksonville's roster.
The team is expected to add at least one tight end in the draft next month.
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