It's not known if talks have reached an advanced stage, or if an agreement is imminent.
Kimbrel, 30, hasn't found a job this offseason despite being one of the most dominant closers of his era. He's coming off a 42-save season for the World Series champion Boston Red Sox that included posting a WHIP below 1.00 for the fifth time and earning his seventh career All-Star selection.
If he were to join the Brewers, Kimbrel would add even more firepower to an already lethal bullpen that features reigning NL Reliever of the Year Josh Hader and former All-Star Corey Knebel at the back end.
However, the Brewers' bullpen could use some insurance in the wake of injuries, as right-hander Jeremy Jeffress will start the season on the injured list with shoulder weakness.
The state of Kimbrel's market remains unclear. The Tampa Bay Rays have apparently maintained contact with the league's active saves leader during spring training. The Washington Nationals and Atlanta Braves, meanwhile, are reportedly no longer involved after previously showing interest.
Signing Kimbrel would cost his new team draft-pick compensation because he declined a qualifying offer from the Red Sox in November.
Milwaukee lost one of its 2019 draft picks earlier this winter after signing free-agent catcher Yasmani Grandal.