Washington feels that Nady or Swisher could improve an offense that finished near the bottom of the National League last year. Who New York would want in return is not known.
Nationals general manager Jim Bowden and team president Stan Kasten were not available for comment.
Nady is a player the Nationals have been interested in since last summer. At the time, the team saw him as a first baseman after it was announced in May that Nick Johnson would miss the rest of the season with a right wrist injury.
Now, Washington sees Nady as an everyday right fielder. Last year, Elijah Dukes played most of the games in right, but he will now compete with Lastings Milledge for the center-field job.
Last year, Nady split time between the Pirates and Yankees and hit a combined .305 with 25 home runs and 97 RBIs. He is arbitration-eligible and will be a free agent after the 2009 season. Nady is currently slated to be New York's everyday right fielder.
As for Swisher, the Nationals would likely use him to replace Johnson. The team feels it cannot rely on Johnson because of his injury problems. Swisher, who was traded from the White Sox to the Yankees in the offseason, hit .219 with 24 home runs and 69 RBIs in 2008.
Swisher was slated to be New York's regular first baseman, but he is now a reserve after the club signed Mark Teixeira to an eight-year, $180 million deal.
The Nationals have tried to improve their offense through free agency, but so far, they have come up short. They tried to sign Teixeira and outfielder Milton Bradley, who signed with the Cubs.
Washington has also tried to sign free agents Adam Dunn and Orlando Hudson, but they have exceeded the team's price range.
Bill Ladson is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.