Bulger's defense team filed the notice Wednesday, The Boston Globe reported.
Bulger was convicted in August on charges for participating in 11 murders, drug trafficking, extortion, money laundering and other crimes while running a criminal enterprise from the 1970s through the 1990s.
U.S. District Judge Denise J. Casper sentenced the 84-year-old to two consecutive life prison terms last week.
Bulger has alleged that he did not receive a fair trial because Casper would not allow him to introduce evidence that former federal prosecutor Jeremiah T. O'Sullivan, who died in 2009, promised him immunity for all of his crimes, including murder, decades ago.
Casper ruled that prior to his trial, Bulger had no evidence of the alleged deal, and that O'Sullivan had no authority to make such a deal in the first place.