The Blue Jays announced the deal Wednesday night.
Travis, 23, spent the 2014 season with the double-A Erie SeaWolves, where he hit .298 with 20 doubles, seven triples, 10 home runs and 52 RBIs. The right-handed hitter posted a .358 on-base percentage and stole 16 bases. He suffered a sports hernia near the end of the season and had surgery in September.
Travis was selected by Detroit in the 13th round of the 2012 first year player draft. Baseball America recently ranked him the No. 1 prospect in the Tigers' organization.
In three minor league seasons, Travis has a career average of .323 with 92 extra base hits in 257 games with 139 RBIs, 41 stolen bases and an OPS of .876.
Gose, 24, batted .226 with a .311 on-base percentage through 94 games with the Blue Jays last season. He had 13 RBIs and 15 stolen bases in 20 attempts.
"Anthony is a true centre-fielder that can provide us with above-average defence in centre field with a good arm," said Tigers president Dave Dombrowski said. "He possesses above-average speed and can steal bases at the major league level. He swings the bat from the left side and we feel that he will continue to improve his offensive game."
The Tigers, in need of outfield depth, had attempted to move Travis to centre field towards the end of the 2014 season.
With files from The Associated Press