JavaScript (Node.js), 90 bytes
JS is clearly not the right tool for this job, but i'm surprised no one posted a JS answer before!
I'm not sure if this is the shortest approach, but this is the best i could come up with :)
For each character of the word, we test if the removal of this character gives a word that is a palindrome, and we retain the index +1 of the last character that succeeded (or -1 if there weren't any).
s=>[...s].map((e,i)=>r=(a=[...s]).splice(i,1)&&a.join``==a.reverse().join``?i+1:r,r=-1)&&r