According to Jon Heyman of MLB Network, discussions between the two sides are "very preliminary." The two sides are expected to meet this week to see if there is anything to talk about, presumably in terms of a possible contract for the veteran.
It is not a secret that the Padres are looking for an upgrade out at shortstop. Alexi Amarista is projected to be the team's starting option on Opening Day. Last year, he hit for a dismal .544 OPS in 357 plate appearances, providing good enough defense to finish out just below replacement level (-0.8 fWAR).
Desmond did not have a great 2015 campaign himself, but he surely is worth a contract for many teams in the infield market. He provides intangibles, such as veteran leadership and postseason experience that could be good for many teams, not just San Diego.
Desmond's market overall has not progressed quite rapidly, but the Giants, Rockies, and White Sox have all been mentioned as possible suitors for him. It's hard to know what type of deal he will get, but MLBTradeRumors.com predicted a five-year, $80 million contract for the shortstop.
The 30-year-old Desmond hit .233/.290/.384 with 19 home runs and 62 RBIs in 641 plate appearances with the Nationals last year. He was much better in the second half than the first, posting a .777 OPS in the back half of the year as compared to a .589 OPS in the front half. According to FanGraphs, he was worth 1.7 WAR.