Burke, who finished last season on loan at Meadow Lane from Aston Villa, has put pen to paper on a two-year contract.
The Irishman scored one goal in seven appearances for the Magpies during the final three months of the 2014/15 campaign.
He made two first-team appearances for Villa and has also had a previous loan spell at Shrewsbury.
The Magpies have also signed Slovenian attacker Filip Valencic.
Valencic, unearthed by the club’s head of recruitment Guy Branston, has put pen to paper on a one-year contract following a trial with the League Two club.
The 23-year-old arrives at Meadow Lane from Italian side Monza where he played six times last season, scoring once.
Valencic, who can play on either wing, has spent the majority of his career in his native Slovenia with Olimpija and Interblock.