Seattle would send Zunino and part-time outfielder Guillermo Heredia to Tampa Bay for 25-year-old outfielder Mallex Smith, according to the Times, which reported that the sides have been discussing a deal since last week.
It wasn't immediately clear if the deal would expand to include more than just the three players.
Zunino, who will turn 28 in March, hit just .201 this past season, but his 20 home runs ranked fifth on the team and came in just 373 at-bats. He has excelled defensively, with 12 defensive runs saved in 2018, ranking first among all catchers and helping him win the Wilson Defensive Player of the Year Award for his position on Wednesday.
Tampa Bay, which traded All-Star catcher Wilson Ramos this past season, has only two catchers on its 40-man roster -- Nick Ciuffo and Michael Perez. Together, they've appeared in just 40 big league games.
Heredia, 27, has been Seattle's fourth outfielder over the past two seasons. In 125 games in 2018, he hit .236 with five home runs, 19 RBIs and 29 runs scored.
Smith was among several surprises for the upstart Rays this past season, hitting .296 with 65 runs scored and 40 RBIs while batting mostly leadoff.
News of the deal came after Yahoo Sports reported this week that Seattle is considering a "full-fledged teardown" this offseason. But Jerry DiPoto attempted to temper that theory at the GM meetings this week in California.
"Clearly overdramatized is the best way to put that," Dipoto told MLB.com, referring to the Yahoo report. "We're open-minded to different ways we can get better, but what we're hoping to achieve is to reimagine our roster to look at it in terms of what is our quickest path to a championship club.
"We know what the Astros, Red Sox, A's, Yankees and Indians look like. We don't want to be a perpetual competitor for the second wild card. We want to build a championship roster. If that means in 2019 we field as competitive a team as we can while earmarking and gathering talent, we're not looking to rip our club down. We're just too talented to do that."