LaRoche has until 5 p.m. ET on Nov. 9 to decide whether to accept the offer.
LaRoche said he would like to return to the Nationals, but he wants a multi-year deal. LaRoche had a mutual option that was worth $10 million, but he turned that down on Thursday morning.
Qualifying offers are a new wrinkle to the offseason. It was created in the Collective Bargaining Agreement last December and replaces the Type A/B free-agent compensation system. In this system, clubs can decide whether to make a qualifying offer to any free agent who spent the full season on that team.
The value of that one-year offer is determined by averaging the top 125 individual salaries from the recently completed season. For 2012, that average is $13.3 million.
By making a qualifying offer, a club is guaranteed a compensation-round Draft pick -- that pick would come between the first and second rounds of the Draft -- if a player rejects the offer and signs with another club in free agency.
The signing team would forfeit its first-round pick, unless that pick was in the top 10. A team with a protected pick (top 10) would surrender its second-highest selection if it signs a free agent who had been given a qualifying offer.
LaRoche is coming off one of his best seasons, hitting .271 with 33 home runs and 100 RBIs. He also won his first Gold Glove Award on Tuesday.
LaRoche said Wednesday that he hadn't talked to the Nationals for about a week and the two parties were at the beginning stages of contract negotiations.
Jackson also wants a multi-year deal but it's doubtful he will get it from the Nationals. If the Nationals are unable to sign Jackson, they will most likely look to acquire another veteran starter.
Jackson appeared in 31 games for Washington and went 10-11 and had a 4.03 ERA. During the second half of the season, Jackson was 4-5 with a 4.78 ERA and lost Game 3 of the National League Division Series against the Cardinals, allowing five runs in six innings.
Bill Ladson is a reporter for MLB.com and writes an MLBlog, All Nats All the time. He also could be found on Twitter @WashingNats. This story was not subject to the approval of Major League Baseball or its clubs.