The two sides have until December 1 to reach a deal, or Swayman will be ineligible to play this season. Meanwhile, the Bruins face uncertainty in goal during training camp.
Joonas Korpisalo, acquired in the Ullmark trade, is expected to step in, but he struggled last season with an .890 save percentage.
Brandon Bussi, who performed well in the AHL, lacks NHL experience. Bruins GM Don Sweeney and head coach Jim Montgomery are expected to address the situation, which remains a focal point as training camp progresses.