SCOTTSDALE, Ariz. — Exactly one month before he is scheduled to make his first regular-season start against the Diamondbacks, Johnny Cueto walked into Scottsdale Stadium on Saturday and declared that despite missing nearly three weeks of camp, he physically is ahead of where he was last spring.

“It wasn’t like I was in the Dominican just sitting down and not doing anything,” Cueto said, with Erwin Higueros translating. “I was working really hard because I knew I was losing time here.”

Cueto flew to Arizona on Friday with his father, Domingo, who is much improved after 10 days in the hospital. Cueto said his dad was diagnosed with a “pre-stroke,” also known as a “mini-stroke” and medically a TIA, or transient ischemic attack.

Domingo Cueto already had kidney issues and one day began acting strangely, not recognizing people and “talking nonsense.”

“It was pretty serious,” the pitcher said. “Since I’m the head of the family, I felt I needed to stay there to make sure nothing worse happened to him.”

Cueto spoke to general manager Bobby Evans, manager Bruce Bochy and pitching coach Dave Righetti before the Feb. 13 spring report date and sought permission to stay in the Dominican Republic rather than come to Arizona and have to fly back if his dad took a bad turn.

The club agreed and placed Cueto on a throwing program. He threw batting practice three times at the Giants’ academy in Boca Chica.

“He looked good,” Bochy said after meeting with Cueto briefly Saturday morning. “He’s excited to be here. We’ll see where he’s at tomorrow. We’re not going to put him in a game right away. He’s going to get a few days to get acclimated here.”

Last spring, the Giants eased Cueto into spring training because he had pitched 237 innings and into late October with the World Series-champion Royals. With more time off this winter, he said, he feels stronger now.

No matter how nonchalant Giants officials sounded about Cueto’s absence, they were relieved to see him in the flesh Saturday, throwing off flat ground and beginning a critical season for him and the team.

Cueto essentially is in a contract year. His six-year, $130 million deal with the Giants allows him to opt out after 2017. If he even approaches his 2016 season, when he won 13 of his first 14 decisions, started the All-Star Game and finished 18-5 with a 2.79 ERA, he almost certainly will declare free agency. At 31, he could command another long and even wealthier contract.

Some pitchers in his shoes might have said no to the World Baseball Classic from the outset to focus on the season, but Cueto was eager to participate. He will skip the first round but said he might pitch in the next round, at San Diego, which begins March 14.

Even that seems quick, considering Cueto will not throw in a game for at least a few days.

Cueto’s 2016 season ended with a 1-0 loss to the Cubs’ Jon Lester in Game 1 of the Division Series. He would have faced Lester again in Game 5, at Wrigley Field, had the Giants not blown a three-run ninth-inning lead in Game 4 at San Francisco.

“I really did want to pitch that fifth game,” he said. “It didn’t happen. The only thing I can think is that this is a new year and you can only focus on what’s ahead of you. I’ve had great success against the Cubs, but you never know what’s going to happen. The Cubs beat us and went on to win the World Series.”

Henry Schulman is a San Francisco Chronicle staff writer. Email: hschulman@sfchronicle.com Twitter: @hankschulman