Bryant suffered the injury in the fourth quarter of last weekend's victory against the Cincinnati Bengals.
The Cowboys believe Bryant will be OK to play wearing a taped-up splint. He was listed as questionable on Friday's injury report.
Bryant met with a team doctor on Monday and a hand specialist on Wednesday and was told his options regarding surgery or possibly playing.
Bryant was told he needs surgery and re-injuring the finger might cause long-term damage. If Bryant wanted to have season-ending surgery, coach Jason Garrett said it wouldn't occur until next week.
Bryant wants to delay the surgery because the Cowboys are in a playoff push. Dallas is just one game out of first place in the NFC East with three games remaining in the regular season.
"It is a mental thing," Bryant said Friday. "I felt like if I go out there and am worrying about my finger, about catching the football, there is no need for me to be out there. I went out (Friday) in practice as if there wasn't anything wrong with my finger. I feel like that's my mindset about it."
Bryant is tied for third in the NFL with nine receiving touchdowns and is 10th in the NFL with 1,028 receiving yards.