Teams had until Friday to make the offers. Players have a week to accept - although none has accepted a qualifying offer in the past three offseasons.
If Zimmermann or Desmond signs with another club before the June amateur draft, the Nationals would receive a draft pick at the end of the first round as compensation.
Zimmermann, 29, is a righty who is 70-50 with a 3.32 ERA in the majors. He was an All-Star twice and threw a no-hitter in the last regular-season game of 2014.
Zimmermann went 13-10 with a 3.66 ERA this season.
Desmond, 30, has a .264 average, with 110 homers and 432 RBIs. He was an All-Star once and won three Silver Slugger Awards.
He struggled for much of 2015, batting .233 with 19 homers and 62 RBIs. Desmond also made an NL-high 27 errors.