Per Heyman, the Padres are "thinking big" and remain "strongly in the mix" for Hosmer's services.
The 28-year-old Hosmer spent the first seven years of his MLB career with the Kansas City Royals before hitting free agency.
He is a 2016 All-Star and four-time Gold Glove Award winner who is coming off his most productive offensive season to date.
Hosmer hit a career-high .318 last season to go along with 25 home runs, 94 RBI and 98 runs scored for a Royals team that finished 80-82.
He also appeared in all 162 games last season for the first time in his career and hasn't missed more than four games in a single campaign since 2014.
Hosmer twice helped Kansas City reach the World Series, and in 2015 he had 17 RBI in the postseason en route to the Royals winning their first championship since 1985.
The Padres are coming off a difficult season that saw them finish 71-91. They performed well beyond expectations, though, as their Pythagorean win-loss figure was just 59-103, per Baseball Reference.
After previously acquiring big names such as Matt Kemp, Justin Upton, Craig Kimbrel and James Shields leading up to the 2015 season, the Padres stripped down their roster and went with almost all young players in 2017.
Their best offensive player was first baseman Wil Myers, who clubbed 30 home runs. If San Diego is able to sign Hosmer, it will likely be forced to move Myers, who came up as an outfield prospect for the Royals, to the outfield on a full-time basis.