Cameron Norrie, Jacob Fearnley, Lloyd Glasspool and Julian Cash are also in the squad.
The winner will face either Australia or Ecuador in the competition’s second qualifying round in September.
In a post on X late last month,, external Draper described the injury as one of the most “challenging” and “complex” of his career.
He said the decision not to compete in the Australian Open, which begins on 18 January, had been “really, really tough”.
Draper struggled with discomfort in the top part of his serving arm – his left – for several months, eventually shutting down his season after withdrawing from the US Open in August.
It came after a successful first part of the year, with Draper reaching a career-high of fourth in the world in June and also securing a maiden Masters 1000 title with victory at Indian Wells in March.