Conklin attended private workout of pre-draft visits with multiple teams, including the Buffalo Bills, San Diego Chargers, Pittsburgh Steelers, Tennessee Titans, New York Jets, Indianapolis Colts, Atlanta Falcons, Detroit Lions, and Tampa Bay Buccaneers during his pre-draft process.

[8] At the conclusion of the pre-draft process, Conklin was projected to be a first round pick and was expected to among the first 15 players selected by NFL draft experts and scouts.

[18] Conklin entered training camp slated as the starting right tackle, replacing Byron Bell.

Head coach Mike Mularkey named Conklin the starting right tackle to begin the regular season.

Conklin joined starting left tackle Taylor Lewan, offensive guards Quinton Spain and Chance Warmack, and center Ben Jones.

[19] He made his professional regular season debut and first NFL start in the Titans' season-opening 25–16 loss against the Minnesota Vikings.

[28] On January 20, 2018, the Titans announced their decision to hire the Houston Texans' defensive coordinator Mike Vrabel as their new head coach.

[35] Conklin started all 16 games for the Titans in 2019, helping Derrick Henry win the NFL rushing yards title.

[38] Conklin was named to the 2020 All-Pro Team, along with three of his teammates: defensive end Myles Garrett and guards Joel Bitonio and Wyatt Teller.

[42] The next day a Week 12 Sunday Night Football matchup against the Ravens, Conklin suffered a right knee injury putting him out of the game.

The next day, on November 29, it was revealed that Conklin tore his patellar tendon and was placed on season ending injured reserve.

[44] In Week 1 of the 2023 season, Conklin suffered a torn ACL and MCL and was placed on season-ending injured reserve on September 12, 2023.