White, who also is nursing a sprained right ankle, pulled his hamstring during Monday night's loss to the New York Jets.
The injury has some in the Falcons organization believing that White could miss multiple games, the sources told Schefter.
The Falcons (1-4) also received bad news regarding Pro Bowl receiver Julio Jones, who will miss the rest of the season with a foot injury. Jones received a second opinion from a foot specialist Wednesday, after Falcons coach Mike Smith said Tuesday that "the first diagnosis was not encouraging."
The team moved quickly to bolster their wide receiver corps, signing veteran Brian Robiskie, son of Falcons wide receiver coach Terry Robiskie, to a one-year deal, multiple league sources told ESPN.
The 6-foot-4, 212-pound Robiskie has the type of size to fill the role of a big receiver. He was a second-round pick of the Cleveland Browns in 2009 following a stellar career at Ohio State. He started 14 games in three seasons with the Browns.
White, who turns 32 next month, has been noticeably limited by injuries this season. The four-time Pro Bowler has just 14 catches for 129 yards and no touchdowns through five games.
Atlanta already is without defensive end Kroy Biermann, fullback Bradie Ewing and linebacker Sean Weatherspoon, who all are on injured reserve. Weatherspoon was placed on short-term IR with a foot injury and is eligible to return in Week 11.
The Falcons have a bye this week before returning to action Oct. 20 against the winless Tampa Bay Buccaneers.
Information from ESPN.com Falcons reporter Vaughn McClure and ESPN NFL Insider Adam Schefter was used in this report.