Injured Twins catcher Mitch Garver has been running some and getting his swings in as he recovers from a mild high ankle sprain suffered May 14, when the Angels’ Shohei Ohtani slid into him.

So far, everything has gone smoothly with Garver’s recovery, making the hopes that he returns sooner than later seem more realistic. High ankle sprains can take four to six weeks to recover from, but the Twins have hoped Garver might be able to back after about three weeks.

“He’s been doing a lot,” manager Rocco Baldelli said. “He’s been moving around. He’s been running. We wanted to get him on the field for an on-field session. I believe he has been hitting in the cage, but we wanted to get him outside. We’ll try not to get ahead of ourselves again, but from what we saw when he originally got hurt to where he is right now, it’s been a pretty quick recovery and I think he’s feeling really good.”

Garver, batting .329 with nine home runs and 19 RBI in 25 games, caught Michael Pineda’s bullpen session Saturday, another good sign. And Twins Chief Baseball Officer Derek Falvey said Garver could be allowed to run the bases during pregame workouts before the homestand ends Tuesday.

The Twins have indicated Garver will need to go out on a rehabilitation assignment before he is activated from the injured list.

Many, many injuries

Falvey met with reporters Saturday to provide injury updates on several prospects. And when the chief baseball officer is providing the injury updates, it means the news isn’t good for the Twins.

Lefthander Stephen Gonsalves was shut down at Class AAA Rochester because of a stress reaction in his left elbow-forearm area. Gonsalves missed the start of the season because of forearm soreness and made his season debut Tuesday, throwing two innings. But the soreness returned. Falvey said the downtime usually is three to four weeks.

Rochester outfielder Luke Raley will miss a few months because of a torn tendon in his left ankle that will require surgery. Raley, who came to the Twins in the Brian Dozier trade in July, is batting .302 with seven home runs for the Red Wings. He was injured May 18 while sliding into second.

Righthander Brusdar Graterol, the top pitching prospect in the organization, has been shut down because of a shoulder impingement. Graterol, with a fastball that reaches 100 miles per hour, left a May 9 start early because of discomfort and eventually flew to the Twin Cities for an evaluation.

Class A Fort Myers outfielder Akil Baddoo will miss the rest of the season because of an elbow injury that will require Tommy John surgery. He joins Miguel Sano and Alex Kirilloff as Twins position prospects who needed the ligament replacement surgery.

In one positive development for the Twins, shortstop Wander Javier was activated at Class A Cedar Rapids after recovering from a strained quadriceps.

In the big leagues …

The Twins are taking their time with designated hitter Nelson Cruz (strained left wrist). He didn’t take any swings Saturday and won’t swing Sunday.

It looks more and more like the Twins have decided to keep Luis Arraez in the majors for now and let Cruz ease his way back into the lineup. The offense hasn’t missed a beat. They have a comfortable lead in the division. Arraez is hitting.

This way, Cruz will be 100 percent when he’s activated.

When asked if he plans to play with some discomfort or try to eliminate all pain, Cruz said: “Just pain free. I don’t want it to be something I’m dealing with all year long. I want to make sure when I go out there it’s 100 percent.”

• Outfielder Max Kepler, who has a sore right knee after colliding with the wall Thursday against the Angels, was given Saturday off after serving as the DH Friday. That put his seven-game hitting streak — during which he is batting .467 with three home runs — on hold.