On Sunday, Newcastle revealed that Armstrong had been left out of their pre-season trip to Ireland as he prepared to depart the newly-promoted Premier League club on loan.
The Magpies claimed that "a number of clubs" wanted to sign the 20-year-old, who has spent the last two seasons on loan with Coventry City and Barnsley respectively, but Bolton have swooped to land the striker until January.
Armstrong told the Championship club's official website: "It's a great feeling to be here. I'm really pleased to have the deal all sorted so quickly and I can't wait to get started."
Bolton manager Phil Parkinson added: "We are delighted to bring in Adam. He's a good young player with great potential and we are looking forward to working with him."
Armstrong made 15 appearances for Newcastle during the 2014-15 season, but he has not been in the plans of Rafael Benitez since the Spaniard arrived at St James's Park.