Remove excess fat from meat. Cut meat into 1-inch pieces. Place meat in a 3 1/2- or 4-quart slow cooker.
Add carrots, celery, mushrooms, and green onions.
Combine in a medium bowl the undrained tomatoes, beef broth, wine, Italian seasoning, salt, pepper, and bay leaf. Pour over meat and vegetables.
Cover and cook on low-heat setting for 9 to 10 hours or on high-heat setting for 4 1/2 to 5 hours.
If using low-heat setting, turn to high-heat setting. Combine in a small bowl the cornstarch and water. Stir into mixture in cooker. Cover and cook for 15 minutes more or until thickened. Remove and discard bay leaf.