Rodriguez was released last week by the Detroit Tigers.
He will report to the Nationals' minor league facility in West Palm Beach, Florida.
After saving 44 games with a 3.83 ERA and 52 strikeouts in 58 1/3 innings pitched last year for the Tigers, K-Rod had a 7.82 ERA and seven saves in 25 1/3 innings this season in Detroit.
The Nationals' bullpen has struggled this season, with an ERA of 4.88, third-worst in the National League, going into Monday night's games. Shawn Kelley, Blake Treinen and Koda Glover all have gotten shots as the team's closer, with Kelley and Glover currently on the disabled list.
Rodriguez, 35, is a six-time All-Star in his 16th season, and his 437 career saves are fourth all-time and the most among active pitchers.
Nationals manager Dusty Baker would not confirm the deal before Monday night's game against the Chicago Cubs but was asked whether he would want Rodriguez for his bullpen.
"It depends if he's still K-Rod," Baker said. "That's what it boils down to. I'll take Sandy Koufax, but he may not be the same Sandy Koufax."
The Associated Press contributed to this report.