This actually made me think.

And read. Negged.
All big cats (except cheetahs) have sharp, retractable claws and elongated canines. Dogs only have the canines.
1) The Dire Wolf was an actual prehistoric canine (dog, not bear-dog), the largest ever and came in at roughly 200 pounds. The largest cat ever (sabre-tooth cat, Smilodon) were pushing 1,000 pounds; they were almost 5x bigger.
2) The strongest bite (accounting for body mass, and in absolute terms) from a big cat (jaguar) is less than the strongest bite from a "big" dog" only when also accounting for mass (African wild dog), but not by much.

3) Also consider that the average AWD is roughly 3 times lighter/smaller than an average sized jaguar.
So if both animals were the same size, the direwolf *might* have the stronger bite, but it probably would not be significantly so. Also, because dogs are omnivores whereas big cats (lions) are hypercarnivores, the lion would most likely have the sharper teeth and thus able to concentrate it's bite over a much smaller surface area (meaning deeper bite).
As much as I hate cats, the direwolf would probably lose. Cats are faster, more agile, have sharper teeth and claws, and can potentially bite much harder. And in the protective mane of a male lion and it's over for the dire wolf.