Brinkley spent five of his first six years with Minnesota after the Vikings drafted him in the fifth round in 2009. He played in 48 games, with 19 starts, over four seasons.
The 29-year-old Brinkley's best season came in 2012, with 100 tackles while starting 15 of 16 games. He was in Arizona in 2013 before returning to Minnesota last season.
Gachkar, who also plays on special teams, played 63 games in four seasons after San Diego drafted him in the seventh round in 2011.
The Cowboys hope to re-sign Rolando McClain after Justin Durant signed with Atlanta and Bruce Carter went to Tampa Bay.
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