Chris Mannix of Sports Illustrated reported the proposal "is believed to be in the neighborhood financially of what he was looking for," though Connelly was seeking a five-year deal.
Connelly started his NBA career with the Wizards as an intern in 1996 and worked his way up the organizational ladder to director of player personnel by 2000.
The Baltimore native left Washington in 2010 to join the New Orleans Hornets as the assistant general manager to Dell Demps. He was hired by the Nuggets as their general manager in 2013 and promoted to his current role overseeing all of the basketball operations four years later.
On Thursday, Candace Buckner of the Washington Post reported becoming the Wizards' president of basketball operations is the "only job" that could lure Connelly away from Denver.
Connelly refused to discuss rumors about joining the Wizards in early April.
"I'm the president of the Denver Nuggets," Connelly told reporters. "Everything else is pure speculation."
He's guided a successful rebuild in Denver as the Nuggets, who bottomed out with 30 wins during the 2014-15 season, have increased their win total in four straight years. They posted a 54-28 mark this year to reach the playoffs for the first time since 2013.
In D.C., he'd be tasked with retooling a roster that currently has the backcourt tandem of John Wall and Bradley Beal signed to max contracts and a limited amount of talent elsewhere.