Medvedev, who won the US Open in 2021, has five titles this year with triumphs in Rotterdam, Doha, Dubai, Miami and Rome. The Russian reached the Wimbledon semi-finals earlier this month, losing to eventual champion Carlos Alcaraz.
Defending Washington champion Kyrgios has missed every Grand Slam this year due to injuries to his knee, foot and wrist. He has only made it onto the court once, losing in straight sets at the Stuttgart Open in June.
The Australian also won the Washington title in 2019 after beating Medvedev in the final.
Japan's Kei Nishikori, Frenchman Gael Monfils and Kevin Anderson of South Africa have been given wildcards into the ATP 500 event, organisers said.
In the women's draw, former world number three Elina Svitolina and 2019 U.S. Open champion Bianca Andreescu lead the list of wildcards.
Ukraine's Svitolina, 28, returned to tennis in April after giving birth to daughter Skai last October and has surged to world number 27 after reaching the French Open quarter-finals and Wimbledon semi-finals.
American Sofia Kenin, who won the Australian Open in 2020, and compatriot Danielle Collins, this year's runner-up in Melbourne, are the other wildcard entries.
