Neal said the Twins are on the hunt for pitching depth inside and outside the organization, and have explored the possibility of adding Giles and the San Diego Padres’ Kirby Yates.
He added that the Blue Jays have been among the teams to send scouts to the most recent homestand of the Twins’ triple-A affiliate in Rochester.
Neal pointed to Minnesota’s top-15 prospects with the Red Wings — Nick Gordon, Brent Rooker and Stephen Gonsalves — as potential trade chips.
Giles, 28, sports a 1.45 ERA with 13 saves and 53 strikeouts in 31 innings with the Blue Jays this season.
The six-year veteran has a 2.68 ERA and 104 saves in his career.