The 24-year-old, seeded ninth, will bid to reach the Wimbledon semi-finals for the first time when he faces Bulgarian Grigor Dimitrov or Britain's Arthur Fery.
Cobolli was defeated by Novak Djokovic in the Wimbledon quarter-finals 12 months ago.
He was also beaten by Alexander Zverev in his maiden Grand Slam final at the French Open in June.