SANTA CLARA, Calif. -- The San Francisco 49ers are still clinging to hope, however small it might be, that Jimmy Garoppolo will start at quarterback Sunday against the Houston Texans.

Garoppolo, who is dealing with a right thumb injury, did not practice again Friday, which means he did not participate in any of the team's four workouts this week.

The Niners are officially listing him as doubtful for Sunday, though coach Kyle Shanahan is not ready yet to name Trey Lance the starter. Lance took all the reps with the starting offense in practice.

"I'm not ready to announce that," Shanahan said. "Jimmy is not out, so I'm holding out hope. He's got a chance this week, so he'll have a better chance next week."

According to Shanahan, the Niners' medical staff made it clear that Garoppolo would have a better chance to heal and potentially play this week the longer he held out from throwing. So, instead of testing the injured thumb in practice, Garoppolo has been held out altogether.

The next step will be for Garoppolo to test the thumb on Saturday or possibly even wait until Sunday morning.

In the meantime, all signs still point to Lance making his second NFL start. Beyond getting all the starting reps in practice, Lance was made available to the media on Friday, his first interview session with local media since after his start against the Arizona Cardinals on Oct. 10.

"Practicing with the first team this week, I don't know if Jimmy is going to be able to go on Sunday, but for me to be able to get those reps, I think, was huge," Lance said. "Whether I can go on Sunday or not, I think I gained a lot from it."

In that start against the Cardinals, a 17-10 loss, Lance finished 15-of-29 for 192 yards and an interception for a passer rating of 58.4. He also had four passes batted or tipped and carried 16 times for 89 yards.

Lance also suffered a sprained left knee in that game, an injury that kept him out and prevented him from continuing to get needed reps in practice. The sprained knee came on the heels of a finger injury suffered in the preseason, both of which helped slow Lance's progress.

But Lance has earned rave reviews from teammates and coaches for his recent work on the scout team, and Shanahan said Lance had a "really good" week this week in preparation for the Texans.

For his part, Lance acknowledged that if he does start Sunday, he will be much further along than he was in October on multiple fronts, crediting Garoppolo and veteran Nate Sudfeld with helping him.

"It was a long time ago that I played," Lance said. "From a health standpoint, I think I feel probably the healthiest I've felt since I've been here. Getting more reps, getting to see things like how Jimmy prepares in a short week, things like that which would have been really hard for me as a rookie, with no vets in the room, I think those things have helped me a ton this year."

In recent weeks, Shanahan has said Lance is playing the best he has played since his arrival over the past month or so, and linebacker Fred Warner has said Lance has been "dicing up" the defense in practice.

"I think Trey's looked good," tight end George Kittle said. "The last couple months, I think that there's been a huge improvement. So, everything you guys saw in camp, I think he's been way better than that. He just looks more mature in the pocket. I think I saw (linebacker) Fred Warner said that he's like not relying on his legs to make plays and he's sitting in the pocket and reading it a little bit more and taking a little bit more chances in the pocket, which I agree with him. ... He's definitely improved in the last several weeks."

With Garoppolo's status still uncertain, the 49ers elevated Sudfeld from the practice squad to the active roster with one of their COVID-19 replacement designations. Sudfeld will back up Lance if Garoppolo is unable to play.

Additionally, running back Elijah Mitchell is listed as questionable against the Texans after missing the past three games with a knee injury. Shanahan said Mitchell had his best day since the injury on Thursday and had no setback on Friday, which would indicate Mitchell should be available so long as he doesn't have any issues before Sunday.

"Hopefully that will continue all the way up to Sunday," Shanahan said. "We'll play Elijah as much as he can handle, so that's about watching him, watching how his legs work, communicating with him, which is really what [running backs coach] Bobby [Turner] does throughout the game, and when he's tired, if he's going, Bobby doesn't hesitate, he puts Jeff [Wilson Jr.] in there and we have a number of guys we can put in there."