While the report doesn't specify the claiming team's league, it seems nearly impossible to think that Dunn could pass through every team in the National League to make it to the American League.
As of today, Dunn should be owed the final $3.91MM of the $12MM he's making in 2010, a price that many contending teams could stomach for the slugger's line of .277/.365/.578 and 28 homers.
As Morosi points out, the claim (while entirely expected) does limit the Nationals' trade negotiations to just the one team. If an agreement can't be reached, he'll be pulled back and the Nationals will keep him through the season's end in order to collect two draft picks (Dunn ranks as a Type A free agent).
Dunn makes sense for several NL West teams, with the Rockies, Giants, and Padres all still in the thick of things. Keep in mind, it's unlikely that the Rox and Giants would allow Dunn to slip to the first-place Padres.