The San Antonio Express-News quoted a source Sunday saying that Parker "absolutely" will be ready to play in Game 1 despite an ankle injury that plagued him for much of the Western Conference finals.
One source told ESPN.com on Monday that the Spurs, while insisting there are "no absolutes," are optimistic that the four full days of rehab Parker is poised to have between Saturday's Game 6 of the Oklahoma City series and Thursday's opener against the Heat will be enough for him to heal.
"That is the hope," the source said.
An official update might not come until Tuesday, when Spurs coach Gregg Popovich is scheduled to meet with reporters for the first time since San Antonio clinched its first back-to-back trips to the Finals in franchise history.
Parker started experiencing soreness in his left ankle in Game 4 against the Thunder. The injury deteriorated to the point that Parker was forced to sit out the second half of San Antonio's eventual series-clinching win in overtime.
How effective he'll be trying to play through the injury remains to be seen. The Spurs have been concerned about Parker's health for much of the season, with Popovich announcing at the All-Star break that he would be cautious with his point guard because of a "variety of maladies."
Popovich said Parker sprained the ankle in Game 4 of the West finals and aggravated it in Game 5. He considered starting Manu Ginobili at point guard in Game 6 against the Thunder because Parker was having trouble during pregame warm-ups. Just before the start of the second half, the decision was made to hold him out.
"He came to me with about eight minutes on the clock [before the second half] and said he couldn't go," Popovich said. "He couldn't cut. He was limping on it. He couldn't cut sideways or forward really."
Popovich said he thought playing Parker in the second half would have hurt the team in the long term.
"I didn't want him to be a hero because he was 50 percent or less, we thought," Popovich said. "That's probably bad for the team, if you think about it. It was a tough decision to try to figure out whether to play him, and if you lose the game and it affects him for Monday night, then you're going to feel like you made an unwise choice."
The Frenchman averaged 16.7 points and 5.7 assists while shooting 50 percent from the field during the regular season, but he missed 22 games. Before the ankle issues, Parker strained his left hamstring in Game 5 of the West semifinals against Portland and did not return, but San Antonio won that game easily to clinch the series.
The Associated Press contributed to this report.