The 29-year-old goalkeeper has arrived as back-up to international colleague Allan McGregor at Ibrox.
"I'm well aware of what is at stake," he said. "But, when Rangers come knocking at your door, you don't turn things like that down.
"It's a fantastic chance for me and my family to come back up and it was too good an opportunity to turn down."
McGregor has established himself as first-choice goalkeeper since Smith's return to Rangers 12 months ago, but Alexander is prepared to battle for the gloves.
"As far as Scotland goes, I know the manager will want goalkeepers who are playing first-team football, so hopefully I can do well enough and we will see where it goes from there," said former Cardiff City goalkeeper Alexander.
"Allan is a fantastic goalkeeper and I am well aware of his capabilities. I have confidence in my own ability and know what I can do."