The 29-year-old has held talks with the MLS club over a potential move to the United States' capital despite the team's concerns about his injury history, according to ESPN FC's Jeff Carlisle.
It's understood that Sturridge has been contemplating offers from up to 13 teams since his contract at Liverpool expired in June.
The Philadelphia Union were also linked with a move for Sturridge, but the club has since moved on, Carlisle added.
Sturridge's potential arrival in D.C. would offset the impending departure of Wayne Rooney, who announced last week that he will leave the club at the end of the season to join Championship outfit Derby County as a player-coach.
D.C. United, however, have reservations given that Sturridge has struggled with a series of hip and hamstring ailments throughout his career.
Sturridge, who spent six years at Liverpool and was part of the squad that won the Champions League in June, is expected to make a decision regarding his future this week, according to Chris Bascombe of The Telegraph.
Along with the United States, he is also reportedly considering offers from Europe and the Middle East. Sturridge has been offered a £9-million contract to join Turkish giants Fenerbahce, Bascombe added.