Neither the Nationals nor Scherzer's agent immediately responded to requests for comment Sunday night.
Scherzer, a 30-year-old right-hander, is a free agent after spending the past five seasons with the Detroit Tigers. He turned down an offer from Detroit last March that would have paid him $144 million from 2015-20.
He was 18-5 with a 3.15 ERA in 2014, a year after going 21-3 with a 2.90 ERA and being voted the best pitcher in the American League.
The NL East champion Nationals' starting rotation already includes Stephen Strasburg, Jordan Zimmermann, Doug Fister -- Scherzer's former teammate with the Tigers -- Gio Gonzalez, and Tanner Roark.