In the last two days, it was learned that managing principal owner Ted Lerner, team president Stan Kasten and acting general manager Mike Rizzo visited Strasburg in Southern California, and all three came away impressed with him.
The Nationals then offered Strasburg a record contract for a pitcher coming out of the Draft.
A source did not give a dollar amount, but it is more than the $10.5 million that right-hander Mark Prior received from the Cubs after the 2001 Draft.
Strasburg, 20, went 13-1 with a 1.32 ERA in 15 starts this season for 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.
If Strasburg doesn't sign, it will be the second consecutive year in which the Nationals couldn't sign their top pick. Last year, right-hander Aaron Crow decided to play for the Fort Worth Cats of the American Association of Independent Professional Baseball after the two sides fell short of a deal. Crow was selected by the Royals in the first round this year, though since he has no college eligibility remaining, he will not be affected by Monday's deadline.
A number of the Nationals would like to see Strasburg signed, though they would understand if management couldn't reach a deal by the deadline.
"I'm on the fence about it, because the organization is doing everything it possibly can, from what I hear, to sign him and not embarrass him or the organization," first baseman Adam Dunn said. "I heard that they already offered him a record contract. How much more does he need?
"It's very frustrating for me, and I can imagine it's frustrating for everybody in here. I can understand how frustrating it is for ownership. I wish they would hurry up and get it done."
Though Dunn believes that Strasburg could make a difference, he also believes there is a risk in signing him. In 1999, for example, the Reds drafted a left-hander named Ty Howington. Dunn went so far as to call Howington the best pitcher he'd ever seen and thought that Howington would play in the Majors, but he never advanced past Double-A because of shoulder problems.
"He was a [six-foot-five] lefty who threw 95 miles per hour," Dunn said. "Personally, he was the best I've ever seen. I knew he would be in the big leagues the next year, at 20 years old, pitch for 20 years and be the best player ever. I really thought that. But there are a lot of things that could happen."
Catcher Josh Bard said that while the Nationals have to do what's best for the organization, there's only so much money the team can give to Strasburg.
"At some point, you have to be reasonable," Bard said. "Obviously, Strasburg is an extremely talented guy, but I could understand where the organization is coming from as far as injuries and all the different things that [could happen].
"I don't know if they are close or far away. I appreciate the fact the Nationals are not trying to negotiate through the media."
Third baseman Ryan Zimmerman, Washington's first-round pick in the 2005 Draft, believes that the Nationals will become a better team once Strasburg is in the fold.
"This organization is going in the right direction," Zimmerman said. "If you want to keep going in the right direction, you sign your top picks. That's how you get better. That's why the Draft is there. That's why the team with the worst record gets the first pick and the best player. Hopefully, he will make the team better, compete and make the playoffs."
Neither Kasten nor Rizzo were available for comment.
Bill Ladson is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.