Adam Schefter of ESPN reported Gould rescinded his contract offers to the Niners, told the front office he was no longer interested in negotiating or signing a long-term deal and asked to get traded.
The 36-year-old Penn State product signed a two-year deal with San Francisco in March 2017. He made 72 of his 76 field-goal attempts (96 percent) and 55 of his 59 extra points (93.2 percent) over the last two years.
Gould, who went undrafted in 2005, made brief stops with the New England Patriots and Baltimore Ravens before settling with the Chicago Bears partway through his rookie season. He ended up sticking in Chicago for the first 11 years of his career.
There was speculation he could end up back with the Bears before the Niners used the franchise tag. In February, he said "once a Bear always a Bear" during an appearance at the Chicago Golf Show (via the Associated Press), but noted he wasn't focused on his future at the time.
"They have the opportunity to get a contract done with me," he said about the 49ers. "Obviously, they have exclusive rights to talk to me until free agency opens. There's no reason not to. ... At this point, I'm just enjoying being around my kids and not really worried about the football contract."
The Bears are still looking for a kicker heading into the 2019 draft, which starts Thursday night in Nashville, Tennessee. Gould's trade demand the week of the draft will likely spark further Chicago rumors.
"The bottom line is, I'm unsure if I want to play [in San Francisco] anymore," Gould told Schefter on Tuesday. "At this point, I have to do what's best for me and my family back home [in Chicago]."
In all, the Pennsylvania native has connected on 97.6 percent of his extra points and 87.7 percent of his field goals in his career. He's tied for third among kickers in Approximate Value since 2005, according to Pro Football Reference.
Gould would make $4.97 million if he signed the one-year tender associated with the franchise tag, per Spotrac. His agent, Brian Mackler, told Schefter the kicker won't report until Sept. 8, the date of the Niners' season-opening game against the Tampa Bay Buccaneers, at the earliest if not traded.