Terms of the offer were not disclosed. The source said getting a deal done "has been a slow process." Published reports indicated that LaRoche would like a three-year deal, but Washington is not willing to go to a third year.
Phone messages to LaRoche and Nationals general manager Mike Rizzo were not returned.
The Nats have had their eyes on LaRoche for several months. He is a solid defensive player and is coming off the best offensive season of his career, having hit .261 with 25 home runs and 100 RBIs for the D-backs.
In early December, LaRoche told MLB.com that he had interest in playing for the Nationals.
"If we come on the same page, I would definitely have interest in playing for them," LaRoche said.
LaRoche said on Sunday he believes the Nationals are going in the right direction, after signing outfielder Jayson Werth to a seven-year, $126 million contract.
"I think it shows how serious the Nationals are about getting the organization back on track and becoming a contender in the National League East," LaRoche said. "They are probably tired of watching other teams [spend money and winning]. With [Nationals Park] and a little money to spend, they are looking to start over."
LaRoche has played in the big leagues for seven seasons. He is best known for playing with the Braves, but he has also spent time with the Red Sox and Pirates. During those seven years, LaRoche was a .271 career hitter with 161 home runs and 569 RBIs.
If they are unable to sign LaRoche, the Nationals will consider Casey Kotchman, an above-average defensive player who is limited with the bat.
This season, Kotchman played for the Mariners, hitting .217 with nine home runs and 51 RBIs. He is best known for his years with the Angels, with whom he hit .296 with 11 home runs and 68 RBIs in 2007.
During his seven-year career, Kotchman has committed just nine errors in 4,875 chances.
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.