Martin Perez discovered a cut fastball that produced a 10-win season in 2019, making him a bargain at the $3.5 million he earned last season.

But doubling that salary? That’s not such a good deal, the Twins decided. They will pay the veteran lefthander a $500,000 buyout rather than trigger his $7.5 million option for 2020. In doing so, the Twins made him a free agent, the fourth member of last season’s five-man rotation to be available on the open market.

Of those four, the Twins made it clear they especially want Jake Odorizzi to return, extending the 2019 All-Star the $17.8 million one-year qualifying offer, which the veteran righthander is expected to decline, preferring to shop for a multiyear contract instead. By making the offer, the Twins would receive a draft pick in compensation should Odorizzi sign elsewhere this winter. Odorizzi has 10 days to accept or decline the offer.

The Twins’ other free agents — pitchers Kyle Gibson, Michael Pineda and Sergio Romo, plus catcher Jason Castro and second baseman Jonathan Schoop — were not extended qualifying offers, and the Twins will receive no compensation if they depart.

Free agency officially began Monday night, and Odorizzi, who enjoyed the best season of his career in 2019, is expected to receive offers from several suitors. The 29-year-old, acquired in a trade with Tampa Bay before the 2018 season, went 15-7 with a 3.51 ERA and struck out a career-high 178 batters in 159 innings.

The list: 2020 free agents by position

Pineda was considered a candidate for the qualifying offer, too, though his next contract comes with an extra hurdle: The righthander must sit out the first 39 games of the 2020 season to fulfill his 60-game suspension for failing a test for performance-enhancing drugs. He cannot return until roughly May 10, costing him about one-fourth of his salary next year.

Like the others, Perez could still re-sign with the Twins for 2020. The lefthander was brilliant early in the season, utilizing the new cutter to produce a 7-1 start and 2.95 ERA in his first 11 appearances. But he faltered as the summer went on, and though he earned an extra $400,000 for reaching innings-based incentives, he posted a 6.29 ERA over his final 21 starts and was left off the Division Series roster.

The Twins also moved on from Kohl Stewart, the No. 4 overall pick in the 2013 draft. The righthander became a free agent when the Twins removed him off their roster. Stewart, 24, has a 4.79 ERA in 17 major league appearances, six of them starts.

And lefthander Stephen Gonsalves, a fourth-round pick in that same 2013 draft, was claimed off waivers by the Mets, the team said. The Twins’ Minor League Pitcher of the Year in 2016 posted a 6.57 ERA in seven big-league games in 2018, then missed nearly all of the 2019 season because of an elbow injury.

The Twins formally picked up designated hitter Nelson Cruz’s $12 million option for 2020, a move they had indicated was coming. They also activated outfielder Byron Buxton and pitcher Sean Poppen from the 60-day injured list, putting the team’s 40-man roster at 33 players.

• After a 101-victory debut season, the Twins’ Rocco Baldelli is a finalist for AL Manager of the Year, which will be named next Tuesday. The other finalists are the Yankees’ Aaron Boone and the Rays’ Kevin Cash. Tom Kelly (1991), Ron Gardenhire (2010) and Paul Molitor (2017) are past winners of AL Manager of the Year, an award that began in 1983.