The British number three has not played a competitive match since retiring from the Barcelona claycourt event in April.
Draper reached the semi-final at Queen's last year, and his absence is a blow to the tournament, which is already missing several big names, including 2025 champion Carlos Alcaraz.
"Recovery is going in the right direction, but I'm going to give myself one more week and aim to return at Eastbourne," the former world number four said in a statement.
"It's very hard to miss one of my favourite events of the year."
Draper's 2025 season was cut short as he nursed a long-term bone bruise in his arm, and he has contested only eight Tour-level matches this year, meaning he has dropped outside the top-100 in the ATP rankings.
The 24-year-old recently added two-time Wimbledon champion Andy Murray to his team.