The 21-year Major League veteran has reportedly been staying in shape in the hopes of hooking on to a team for Opening Day, but has yet to respond to the Royals offer.
Now 40 years old, Pudge played in a career-low 44 games last season due to injury, but averaged 122 games played the five years prior. The former MVP and 13-time Gold Glove winner is just 156 hits shy of 3,000.
The Royals had announced yesterday they were on the lookout for an extra catcher just hours after their projected catcher, Salvador Perez, underwent surgery to repair a torn meniscus in his left knee.
If Rodriguez declines the offer, Kansas City's only other choice of catcher on the free agent market would be former Met and White Sock Ramon Castro.