How should I connect GFCI outlets where the wires loop over the existing outlet?
Option 1, which has the merit of being easy:  Put them both on LINE, just like that. Leave the warning tape on the LOAD terminals. 

This will only provide GFCI protection at this receptacle location, and you would need to buy GFCIs for each socket. 

Now if you want one GFCI device to protect every receptacle on the circuit, you have several options. 

Option 2: Fit a GFCI+breaker

These critters are more expensive than a GFCI+receptacle device, and you need to open up your service panel to do it. But voila, your entire circuit is protected and done.  You could protect 5 circuits in about 10 minutes' work, instead of all the chasing and troubleshooting I'll next describe:

Option 3: Reverse-engineer the circuit to place the GFCI correctly

Every GFCI has a pair of LOAD terminals, which protect loads downstream from the GFCI device itself. That is simple on breakers and plain GFCI (deadfronts). But on GFCI+receptacles, their dual function makes it very easy to misapply the LOAD terminals, and cause big problems.  Hence the warning tape. Heed it. 

You will need to get some testing equipment (dollar-store night lights are great) to detect power on each receptacle socket.   First, turn off the breaker to that circuit, and check absolutely everything to figure out which loads are on that circuit, and which are not.  You don't want to miss a load.  If the breaker is a 2-pole breaker, you have a MWBC and you'll need a lot more skill to do this; instead just punt back to the "Fit a GFCI+Breaker" option using a 2-pole GFCI breaker. 

Take a guess at which receptacle load is the first one reached by the cable from the panel.  Regrettably, you'll need to cut that nifty loop there.  If this turns out to be the wrong one, darn.  Strip a little more insulation back so the wires can go into a wire-nut or around a screw terminal, and either sit there making 4 shepherd's hooks and putting them on 4 screw terminals, or just get a 6" length of black and white wire and pigtail the receptacle using wire-nuts.  My knees prefer the latter. 

If you found the right location, when you sever those wires (cap them off temporarily with wire-nuts), the entire rest of the circuit will be dead when you turn the breaker back on.  That is the best place to put the GFCI+receptacle device.  Now you need to figure out which black/white pair is the supply.  Trial and error will suffice.  Attach that to the LINE terminals as I describe here: 

Best way to use LOAD terminals without it going badly

STEP 0: Grounds go everywhere. 

The GFCI engine doesn't use ground.  But they are often packaged in  or combined with things which do need ground.  (e.g. a GFCI+receptacle combo device, whose sockets need ground).  So if able, ground it. Grounds are always pigtailed so you can remove the device without severing the ground. 

STEP 1: Attach only 2 conductors to the LINE terminals.

Over the course of mapping the circuit, you will find 1 cable/wire pair (black, white) that you think may be your supply hots.  

Connect only that pair only to the LINE terminals of the GFCI.  (obviously also connect ground).

Give the GFCI a full shakedown cruise.  Make sure it powers up, make sure a big load like a toaster works on it, try its test/reset buttons, try a 3-lamp tester/external GFCI tester (ground needs to be hooked up for an external GFCI tester to work) etc.  Until you do this, do not pass GO, do not collect $200. 

STEP 2: Hook up anything else to the LOAD terminals

Without disturbing work so far, only now tear the warning label off the LOAD terminals, and to them, hook up any additional (downline) loads.  If you need to land 2 on there, use a pigtail.  

If you get any GFCI trips that you aren't intentionally causing via testing, then separate all your loads and apply one load at a time until you find the load that is tripping.  That load has a problem. The problem is probably not a defective GFCI device, it's a defective load that needs cleaning, repair or replacement.  Do that. 

Then label each downline receptacle "GFCI Equipped" and if applicable, "No Equipment Ground".