Strasburg went 13-1 this season for the Aztecs, leading Division I pitchers with a 1.32 ERA and 195 strikeouts. Last month, he won the Golden Spikes Award as the top amateur player in the United States.
If Washington can't reach an agreement Monday with the right-handed pitcher from San Diego State, the team will lose its rights to him.
Nationals president Stan Kasten told The Associated Press over the weekend that he thought a deal might not get done, even though Washington offered Strasburg a contract that would be the biggest ever given to a player taken in baseball's amateur draft.
The record for a rookie signing was the $10.5 million the Cubs offered to Mark Prior in 2001.
MASNSports.com, a Web site co-owned by the Nationals and the Baltimore Orioles that focuses on sports in the mid-Atlantic area, reported that "sources close to the negotiations" said the Nats have put together a deal offering $17 million, plus $3 million in incentives.
Scott Boras, who is representing Strasburg in the negotiations, told The AP that "for the first time in eight years, we have an extraordinary player who is in the draft marketplace. Wouldn't we all like to buy bread at a price they charged 100 years ago, a nickel a loaf, with the income we enjoy today?"