Here's what the wiki tells us:
It's possible — but not decided yet. The Colts fired GM Chris Ballard after a 2-4 start, creating major uncertainty. Multiple teams have expressed interest in Jonathan Taylor ahead of the November 10 deadline (Vikings and Steelers are mentioned as potential landing spots). (from the Jonathan Taylor page)
Steichen's role in this is complicated. He's on the hot seat himself — PFF named him one of six NFL head coaches facing pressure, with a 25-26 record and zero playoff appearances in three years. If the new GM leans toward a full teardown/rebuild, trading Taylor (their most valuable asset) would free significant cap space. But Taylor's August extension ($44M over two years, $39M guaranteed) makes any deal trickier — a contender would need to absorb that cap hit. (from the Shane Steichen page and Jonathan Taylor page)
The wild card is who replaces Ballard. A new GM could go either way: keep Taylor as the centerpiece of a quick turnaround, or ship him for picks and start fresh. Steichen's own job likely hangs on that decision too.
Underdog's Josh & Hayden noted earlier in the season that Taylor's usage under Steichen has been solid — "RB3 overall usage over the past three seasons when Steichen has arrived... he's had some low-end RB1 moments and then true elite RB1 moments." So the coaching staff has leaned on him. But with the franchise direction up in the air, nothing is locked in.
Bottom line: Trade chatter is real, the team is spiraling, and multiple suitors are circling. A Taylor trade before Nov. 10 is a live possibility, but it'll ultimately depend on what the new GM decides — and whether Steichen survives the shakeup long enough to have a say.