In Post Crisis, Clark was chosen by the ring first. But Abin Sur did not want a Kryptonian to have the ring, he wanted a human. So, the ring went out again and snagged Hal. In modern times, however, we don't know if this happened or not. It has neither been confirmed nor denied. So, we don't know. Since the modern Superman and Hal has kept most of their Post Crisis history, it's safe to assume this did happen. But again, we don't know for absolute certainty.
Side Note: Back in the late 80's, early 90's, Hal was dying and a number of successors were chosen to replace him. If I remember correctly, I think John Stewart and Guy were among that number (this was before either of them became GL's). But Clark Kent was also among that number as a potential replacement as well. So, that's twice that Clark could have potentially become a Green Lantern. Hal would dismiss the potential replacements and willed himself to stay alive. He was successful.
Side Note 2: Back when Superman lost his powers and the comics jumped to one year later, Hal offered Clark a power ring. Kent took it, used it to float up in the air but then returned it, saying he didn't wish to come out of retirement. So, once again, Superman could have become a GL but this time he outright refused. Geoff Johns wrote this story and I really wish he had explored Superman, who had no powers at that time, with a power ring. It would have been a neat story where Superman used the power ring to fight crime and what he could do with it. We also know it would have been temporary as Superman's powers would have returned and Clark would return the ring to Hal. But no, Johns didn't explore this. Talk about a missed opportunity. A huge one.