"The bottom line is, I'm unsure if I want to play there anymore," Gould told ESPN. "At this point, I have to do what's best for me and my family back home."
The veteran kicker's decision comes a day after general manager John Lynch expressed hope to keep Gould, who received the franchise designation in February, around for the long term.
"Robbie's going to be a part of us this coming year, I know that," Lynch told reporters Monday, via a team transcript. "We would like it to be longer than that. We've made an attempt to make that happen.
"We haven't come to an agreement as of yet and we'll see where that goes, but Robbie will be a part of us this coming year. We're excited for that because he's very good at what he does and he's also a big part of this team."
Gould joined San Francisco on a two-year deal in 2017 leading to the franchise tag, and the two sides have until July 15 to work out a multiyear deal.
The kicker's request to be traded almost certainly signals he won't be signing the franchise tag anytime soon even if the Niners don't ship Gould elsewhere.
"At this time, we are unsure when or if he will report," Gould's agent told ESPN. "It will not be prior to Sept. 8, at the earliest, if at all."
Either way, the Niners better start exploring options (perhaps a return to Chicago for Gould) as the offseason progresses.