No, unfortunately.
I suppose I can see that working – Git generates a temporary file based on what’s currently in the index, hands it to the difftool along with a copy of the current work tree version (to protect you from making further changes), lets you use the difftool to move some of the changes to the index version, then once you save and quit, stages whatever content is in that modified index version. Note that this would require the difftool to also be a bit of an editor, and not all valid difftools are; some of them are just for viewing diffs. Note also that this is basically bypassing all of git add -p. You wouldn’t have any of the normal interface from it for moving between hunks, splitting hunks, and so on. The difftool would be entirely responsible for all of that.
If your difftool is fully-featured enough to do this sort of thing, then I suppose you could write a script to do it. An outline, without really any error protection, handling of special cases (binary files?), and completely untested:
#!/bin/bash
tmpdir=$(mktemp -d)
git diff --name-only |
while read file; do
cp "$file" $tmpdir
# this has your changes in it
work_tree_version="$tmpdir/$file"
# this has the pristine version
index_version=$(git checkout-index --temp "$file")
# and now you bring changes from the work tree version into the index version,
# within the difftool, and save the index version and quit when done
my_difftool "$work_tree_version" "$index_version"
# swap files around to run git add
mv "$file" "$work_tree_version"
mv "$index_version" "$file"
git add "$file"
mv "$work_tree_version" "$file"
# you could also do this by calculating the diff and applying it directly to the index
# git diff --no-index -- "$file" "$original_index_version" | git apply --cached
rm -r $tmpdir
Probably a lot of ways to improve that; sorry I don’t have time to be careful and thorough with it right now.