Something Real,
The option for Sony/Tristar to make another Godzilla film expired in 2003. They only own television and home video rights for the '98 movie in perpetuity. They also own the North American home video and television rights for the Heisei movies and Millennium films. They'll have the option to re-new those rights around 2022, give or take a couple years. (Sans The Return of Godzilla and Godzilla vs. Biollante.)
So with their option to make another film with the title or character "Godzilla" long expired, and Toho's current deal with Legendary Pictures in full swing, Sony has no more rights to make a new Godzilla movie than you or I do.
EDIT: Correction and more specifics. Sony's television and home video rights for Godzilla vs. King Ghidorah and Godzilla and Mothra: The Battle for Earth end in 2019. The rest of the Heisei and Millennium movies they own North American distribution rights to end 2023-2025.
"'Nostalgic' does not equal 'good,' and 'standards' does not equal 'elitism.'" "Being offended is inevitable. Living offended is your choice."