The Cowboys are trying to gather as much information as possible regarding Bryant's finger to determine whether he needs season-ending surgery or can continue playing using tape and a splint.
If Bryant reinjures the finger, it could cause permanent stiffness, which would minimize his ability to catch passes.
"One of the things that we know about Dez is that he's a tough guy and he loves playing football," Cowboys coach Jason Garrett said Wednesday morning. "He's going to give himself and our team every chance possible to play in (Sunday's game against the Pittsburgh Steelers) and continue to play this season.
"We just need to get more medical information about how to handle the situation right now, the best way to handle it for now and for the future."
Bryant initially saw a hand specialist Monday, when the fracture was diagnosed.
It remains unclear who ultimately will decide whether Bryant needs surgery or plays. A source close to Bryant told ESPN that the receiver will play against Pittsburgh if he receives medical clearance.
Garrett said Monday that it was a medical decision regarding Bryant's playing status. But one day later, Cowboys owner Jerry Jones said on his weekly radio show on KRLD-FM that the decision was up to Bryant.
Jones used an example of how former Cowboys receiver Terrell Owens faced a similar situation in 2006. Owens fractured a finger but elected to keep playing instead of having surgery. He didn't reinjure the finger and had surgery at the end of that season.
"I think like with any decision, there are a lot of people who are involved in the decision," Garrett said. "Certainly the player has a lot to do with it -- his willingness and his ability to play.
"Again, the medical part of it is a big piece and his ability to function is a big piece. So we'll continue to evaluate that today and get some more information today and try to make our best decision going forward."
Bryant has been one of the Cowboys' best performers this season. He's caught 30 passes for 438 yards with six touchdowns over the past four weeks, and he's scored in the past five games.
Bryant ranks 10th in the NFL in receiving yards (1,028) this season and is tied for third with nine touchdowns.
ESPN's Josina Anderson contributed to this report.