Naturally, no.
In history the biggest birds that still could fly were the Haast's eagles with a wingspan of 3m, and still those flying giants only weighted 15Kg and couldn't carry much. If we go farther back we have Quetzalcoatlus, the largest flying animal, with a wingspan of 15m and a estimated weight of 70Kg.
So, having the size of a lion is no problem. And in the case of Haast's eagle it had 6cm claws and was a powerful predator.
So, what you need to create this gryphon-like animal is a similar environment to the one that created the Haast's eagle: Isolated island with a lack of land predators that would have allowed birds to fill the niche.
So now you have a large bird apex predator, it flies and could have developed some feline-like characteristics via convergent evolution. Locals could have learned to tame it to send small packages and messages across the islands.
But, to go further we will need to play around with some more magical elements.
Flying is hard, it requires an animal to specialize, reshape it's whole body and metabolism for this single task... But, what if it wasn't?
Let's say we introduce cavorite in this bird's diet, the mineral could be naturally occurring in higher altitudes and end up accumulating in the bones. Now these birds get some form of naturally occurring anti-gravitational abilities that make flying so much easier!
Without the constraints of weight they can grow and develop muscles that are used for more than just flying. Now they can grow big enough to carry riders and to prey on animals the size of elephants.
The last burden to overcome are the six-limbs, unless you want to make those gryphons actually some weird form of insect, they would need to develop some really lucky mutation that would give them an extra pair of limbs.