Morton, who turns 37 next month, was due to earn $15 million. Zunino's option was for $4.5 million.
Both played key roles in the Rays reaching the World Series, where Tampa Bay lost to the Los Angeles Dodgers in six games.
General manager Erik Neander had until Sunday to decide whether to exercise the options. He spoke with both players Thursday night.
"With both of them, the door is staying open. We're going to continue talking," Neander said Friday.
"Charlie's been very open about the uncertainty in his future and where he's at," Neander added. "Frankly, I felt it was best to let him catch his breath, let him process, let him make some decisions with his family and have those conversations to give us a little more time. I would love nothing more than to have Charlie back as part of our club next year."
Neander said he also had a good conversation with Zunino, who started 18 of 20 postseason games, including the final 10.
"The intangible benefits of him ... and the work he did behind the plate were a huge reason for our success as well," Neander said.
Morton joined the Rays in 2019 on two-year, $30 million contract and helped them earn playoff berths each of the past two seasons. The right-hander was 3-1 with a 2.70 ERA this postseason, with the loss coming in Game 3 of the World Series. He was 2-2 with a 4.74 ERA during the season and earned a $5,555,556 prorated salary from an originally scheduled $15 million.
The 29-year-old Zunino was obtained in a trade from Seattle in November 2018. He batted .170 with four homers and eight RBIs in 19 postseason games this year.
Zunino batted .147 with four homers and 10 RBIs during the season and earned $1,666,667 in prorated pay from a salary that had been $4.5 million.