Luring the behaviour
Ask your puppy to sit, hold the treat between your fingers, and hold it close to your nose, now lower your hand slowly down to the floor directly between your puppy's front paws. Hang on to the treat by turning your palm down, with the food held within your fist. Be patient and wait, eventually, your puppy should flop to the floor. When your puppy is in the down position click or say 'Yes' then reward with the treat. (Feed the treat on the floor for this, as it prevents your puppy from following your hand back up again like a yo yo) Repeat 10 times.
After 10 successful repetitions do the same as above but this time you will fake having a treat in your hand, so do the same hand action and if your puppy lays down you will click, or say 'Yes' and reward. Repeat this step 10 times.
Adding the verbal cue
After 10 successful repetitions, add the verbal ‘Down’ just before moving your hand as before. The sequence should look like this,
If the above version did not work, then try this, form a small low bridge with your leg, using a treat to lure your puppy through and under your leg, the bridge should allow them to crawl underneath and take the down position. Don't put any pressure on their backs. Once they are in the down position, click or say 'Yes' and reward. Repeat 5 times, before trying the above version again.