Burgos and Stites have both seen time with the D-Backs this season. Drury, however, has yet to get a taste of the big leagues. The 21-year-old was a 13th-round draft pick by the Braves back in 2010.
He is making his Major League debut on Tuesday, as he is the starting third baseman for Game 1 of the doubleheader against the Rockies.
Escobar had not played since leaving Thursday's game with a hand injury following a hit-by-pitch.
Gosselin is in Monday's starting lineup against the Rockies. He is batting seventh while playing second base.
Meanwhile, infielder Yunel Escobar remains out with a hand injury. It will be the third straight game he has missed due to the issue.
Gosselin has not played since mid-May after undergoing thumb surgery. He was acquired from the Braves in June.
Both players were injured during Thursday's game and will be missing their second straight game with their injuries. The Nationals have not passed along any updates on their status, so they are both considered day-to-day at this point.
Anthony Rendon will assume his position with Trea Turner playing second.
Escobar had been on a roll before going down with nine hits in his last 19 at-bats to sweeten his slash line to .308/.367/.423.
He remained on the bases, but left when his team took the field in the bottom of the inning.
X-rays taken of Escobar's hand came back negative, and he is considered day-to-day for now.