A baseball source did not give a dollar amount, but it is more than the $10.5 million right-hander Mark Prior received from the Cubs after the 2001 Draft.
Team president Stan Kasten did not return a phone message and acting general manager Mike Rizzo, who is on the road trip with the Nationals, declined to comment on the Strasburg situation.
The source said Washington did not come away believing that it would reach a deal with the right-hander before the midnight ET deadline on Monday. In fact, Kasten was realistic and told The Associated Press that Strasburg may not sign.
"With 48 hours to go, I simply have no idea whether we're going to be able to reach a deal," Kasten told the AP.
The Nationals feel strongly that Strasburg can make an immediate impact in their rotation, maybe joining the team's Major League roster sometime in September.
If he is promoted to the big leagues this season, Strasburg will become only the third pitcher in baseball history to be drafted No. 1 and play in the Majors in the same year, joining Rangers left-hander David Clyde in 1973 and Orioles right-hander Ben McDonald in '89.
Strasburg, 20, went 13-1 with a 1.32 ERA in 15 starts this season at San Diego State University en route to being named the Golden Spikes Award winner. He struck out 195 batters and issued just 19 walks in 109 innings.
"This kid is so impressive. We have nothing but the highest regard for him. If he wants to come and begin his career right now -- and do so with the largest contract ever given to any drafted player in the history of Major League Baseball -- we can help him accomplish that," Kasten told the AP. "But if this is more about changing the whole way an industry does business, then we won't be able to reach a deal.
"We think he's not just a Draft pick -- we think he's a special player, which is why we extended ourselves as much as we did, and will continue to entertain any other issues or concerns he might have in order to get him signed."
Bill Ladson is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.