Konerko will be in his 18th season and began his career with the Los Angeles Dodgers before being traded to the Cincinnati Reds.
The deal is reportedly a one-year, $2.5 million contract.
Konerko, 37, hit .244 with 12 homers and 54 RBIs in 126 games last season, playing through thumb and wrist injuries.
The Colorado Rockies are close to a two-year contract with free-agent first baseman Justin Morneau, MLB.com reported.
Multiple media outlets reported that the potential deal is worth $13 million.
The Rockies added some flexibility to their payroll when they traded outfielder Dexter Fowler and his $7.35 million salary to the Houston Astros earlier in the day.
Morneau, 32, is expected to take over as the Rockies' starting first baseman with Todd Helton retiring.
For his career, Morneau is a .277 hitter with 221 home runs and 863 RBIs. He was the American League's Most Valuable Player in 2006.
The New York Yankees are closing in on a deal with free agent second baseman Kelly Johnson, according to multiple reports.
The team is reportedly looking at Johnson as an insurance policy in the event it cannot reach a deal to re-sign prized free agent Robinson Cano.
Johnson, 31, can also play third base and outfield. He batted .235 with 16 home runs and a .715 OPS with the Tampa Bay Rays this year.
Major League Baseball announced that New York Yankees minor league catcher Ryan Baker has been suspended 50 games without pay for refusing to take an offseason drug test, which violates the minor league drug prevention and treatment program.
Baker, who is on the Class A Charleston RiverDogs' roster, will begin his suspension at the start of the 2014 season.