Colorado catcher Tony Wolters also could be headed toward a hearing next month after requesting a raise from $960,000 to $2,475,000. The Rockies offered $1.9 million.
The club agreed to one-year deals with its four other players who were eligible for arbitration. Right-hander Jon Gray received a raise from $2,925,000 to $5.6 million, left-hander Kyle Freeland from $565,000 to $2,875,000, outfielder David Dahl from $584,288 to $2,425,000, and reliever Carlos Estevez from $565,000 to $1.08 million.
Right-handed reliever Scott Oberg also was eligible for arbitration before reaching a $13 million, three-year contract with the Rockies last month.
The 27-year-old Story is coming off a season in which he set career highs in runs (111), batting average (.294) and on-base percentage (.363) while earning $5 million.
Story finished with 35 homers and 23 steals, making him the second shortstop in major league history with multiple 30-20 seasons. Alex Rodriguez is the other one.
It also was Story's fourth straight season with at least 20 homers. He's the first shortstop in big league history to begin his career with four consecutive 20-homer seasons.
Wolters set career highs in games played (121), batting average (.262), runs (42), hits (94), doubles (17) and RBIs (42). He appeared in 112 games at catcher, tied with Yorvit Torrealba (2007) for second-most in team history.
Wolters, 27, lined his only homer of the season on May 22 in Pittsburgh. It snapped a streak of 247 at-bats without a long ball. It was his first homer since Aug. 27, 2018.
Gray had a solid season until being placed on the 60-day injured list Aug. 21 with a broken left foot that required surgery. The 28-year-old went 11-8 with a 3.84 ERA over 26 games (25 starts). Gray was the third overall pick in the 2013 amateur draft.
Dahl hit .302 with 15 homers and made his first All-Star team in an injury-plagued season. He spent time on the injured list twice in 2019 with an abdominal strain (April 9-18) and then a right ankle sprain (Aug. 3 to the end of the season).
Dahl, 25, was the 10th overall pick in the 2012 draft. He batted .367 last season with two outs -- the highest mark in the NL.
Freeland, a Denver native, was the team's opening day starter. Freeland never really found his command and was even sent to the minors for some fine-tuning.
The 26-year-old Freeland made a career-low 22 starts. He was 3-11 with a 6.73 ERA. He also found himself on the injured list twice, once with a blister on his middle finger (April 22-29) and then with a strained groin (Aug. 2 to Sept. 21).
Big things were expected last season from the eighth overall pick in 2014. Freeland, who went 17-7 with a 2.85 ERA in 2018, couldn't get on track. He was optioned to Triple-A Albuquerque on May 31. He went 0-4 with an 8.80 ERA for the Isotopes before being recalled on July 13.
Estevez appeared in a team-high 71 games. His 72 innings out of the bullpen were tied for 10th in the league. The 27-year-old was 2-2 with a 3.75 ERA.
Estevez began the season with Triple-A Albuquerque before being recalled on April 2. He spent most of the 2018 season on the injured list because of an oblique strain and right elbow strain.