Burke was injured in the first quarter of Utah's 106-74 loss to the Los Angeles Clippers on Saturday night at EnergySolutions Arena.
Afterward, Lucas talked bravely of accepting more responsibility from coach Tyrone Corbin.
"I'm just going to play," he said. "I'm going to come in and hold it down. My whole thing is to come in and play right away. Whatever Ty wants out of me, that's what I'm going to do, no matter what.
"I live for this game and, when the opportunity comes, I'm going to step it up. I ain't shying away from nothing. I'm just going to play hard, play tough and play with a lot of energy."
Burke will be examined by a hand specialist Monday, when a decision on possible surgery will be made and his official time-frame for returning will be announced.
Burke's father, Benji, is also his agent.
On Sunday afternoon, he sent The Tribune a text message that read, "Not sure on surgery. Looking at three to six weeks, guessing. [Trey] is in great spirits."
The Jazz selected Burke in the first round of last summer's draft.
He was expected to quickly become a foundation player for the rebuilding franchise.
"My heart goes out to him," Lucas said, "because he wants to be out there and he was just now getting into his rhythm. He was feeling comfortable, feeling the flow of the game how fast [it] can be, the intensity. [But] one thing about Trey, he's going to go in and watch film. And when he comes back, he's going to be ready."
After the loss to the Clippers, Corbin spoke of management "putting our heads together" and deciding what to do with the roster while Burke is sidelined.
Currently, the healthy point guards in camp are free agents Scott Machado and Lester Hudson, along with Lucas.
"It's the NBA, man," Corbin said. "You've got to adjust. We're working a lot of things out. We knew we'd have to. This is another obstacle."
Like Corbin, Jazz players expressed confidence in Lucas, a 30-year-old veteran who has started two games in his five-year NBA career.
"He's been a great locker room guy," said Derrick Favors. "He comes in, even during practice, and gets everybody ready. He's very vocal and he's a great guy a great teammate. I'm glad he's on the team."
Burke, talking to reporters with a large splint bandaged to his injured finger, also endorsed Lucas:
"He brings, first and foremost, leadership. He's played pretty much everywhere. He's able to knock down shots. He's able to get guys going and get guys in the right spots. We have no doubts he can do it. We all know what John brings to the table."
Trail Blazers at Jazz
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