Burton was reportedly unhappy after Cameron Ciraldo shifted him into the centres mid-season, swapping places with Bulldogs captain Stephen Crichton.
He still considers himself a five-eighth, a position that pays considerably more than centre.
Chiefs, Bears, Parramatta and former club Penrith are among the clubs believed to be interested in him.
Burton officially becomes a free agent next weekend, but has sought permission to speak with rival clubs early.
Canterbury football boss Phil Gould confirmed the Bulldogs granted the request.
Gould wrote on X: "Matt's manager has requested permission to test the market… it's a good way for both Matt and the Bulldogs to gauge his true value and give both parties more information,"
He posted Friday night and stressed the Bulldogs "at all times retain last rights," presumably only until November 1.
After his first post drew speculation and cynicism, Gould clarified: "Just for the record we love Matt Burton, we want to keep him, Matt Burton loves the Bulldogs, he wants to stay,"
He added: "Matt's management doesn't want Matt underpaid, Bulldogs don't want Matt overpaid… that's both parties being responsible.
"Simple solution, let the market tell us what the true value might be, then we can have a more informed discussion."
Gould suggested neither side had yet discussed Burton's pay for the 2028 season and beyond.