The Tractor Boys have been looking for a successor to Mick McCarthy since April and are now thought to be on the verge of appointing Hurst, who took his Shrewsbury side to the League One playoff final this season.
Shrewsbury's unlikely promotion push ultimately ended in heartbreak, but Hurst's performance in charge of the club is understood to have impressed the Ipswich board.
"Town have been given permission to speak to Shrewsbury boss Paul Hurst over the vacant manager's position at Portman Road," read a statement on the club's official website.
"Blues have agreed a compensation package with the League One club but personal terms are still to be finalised."
Hurst took over the Shrews in October 2016 when the club were bottom of League One, saving them from relegation before almost leading them to the Championship this term.
The 43-year-old is now expected to take over an Ipswich side that finished 12th in the Championship last season.