The facility is independent of the 49ers, a requirement important to the NFLPA because it wants to ensure that Smith is treated as a person and not as a player.
Fox Sports earlier reported on Smith's treatment being independent of the 49ers.
The union doesn't want Smith's return date to the 49ers to be a factor in determining when he will be finished with treatment.
"It's not like he's going to be magically cured in a month," a source close to Smith told ESPN. "If his problem is serious enough, a month might not be enough."
The source told ESPN that Smith's top priority should be his recovery, not his return to the 49ers.
"The only timetable on this is him getting well," the source said. "I know football fans don't care about this, but he's 23 years old and he has a full life beyond football. How he recovers from this can literally be the difference between life and death."
Smith, who turns 24 on Wednesday, has been arrested twice for suspicion of driving under the influence. He received required counseling for alcohol abuse following his first DUI.
The 49ers placed Smith on the non-football illness list Monday, three days after he was arrested again.
If convicted, he faces an automatic two-game suspension for multiple violations of the league's substance-abuse policy
After playing in Sunday's 27-7 loss to the Indianapolis Colts, Smith apologized to his teammates, and team CEO Jed York said the linebacker was leaving the 49ers indefinitely.
"I am taking a leave of absence to address my health," Smith said in a statement released by the team. "I am sorry that I have affected my team, my family and the organization. I will do everything in my power to handle this situation the best way possible. I appreciate the support of the 49ers and our fans."
An NFL spokesman said there is no minimum or maximum time Smith can be on the non-injury list. ESPN NFL Insider Adam Schefter reported Monday that Smith is expected to be away from the team for about a month.
Information from ESPN's Adam Schefter and ESPN.com 49ers reporter Bill Williamson was used in this report.