Huddersfield possess the Premier League's fourth worst defence this season, leaving them third from bottom of the table.
Heaton, meanwhile, has fallen down the pecking order at Burnley, after Joe Hart's arrival from Manchester City in the summer transfer window.
And the Sun claims the three-cap England international is desperate to leave Turf Moor, where another Three Lions shot-stopper, Nick Pope, is also employed.
But do the Huddersfield fans want him? This is what they are saying on Twitter...
Is he now a striker..?— Nevo (@HTAFCYork) November 9, 2018
...we don't need another bloody keeper!— Lee Robinson (@bilbobarneybobs) November 9, 2018
No we don't!— Lee Oates (@LeeAirborne) November 9, 2018
But not everyone is against the idea - in fact, far from it.
Goalkeeper is a big issue.— HuddersMike (@Huddersmike) November 9, 2018
Id have him im a heartbeat, especially as likely on a loan anyway, lossl not particularly done anything wrong but not exactly setting league alight, cant see a rush for him if we went down.— Watson (@dpw247) November 9, 2018
Absolutely brilliant move if it comes off.— Karim Fatih (@karimfatih) November 9, 2018
Unquestionably better than Lossl.— Karim Fatih (@karimfatih) November 9, 2018
Heaton's Burnley contract only has 18 months left to run, and the presence of both Hart and Pope could tempt the Clarets into selling the 32-year-old.
A move to Huddersfield, who are based just 25 miles east of Burnley, could appeal to Heaton, whose roots are in the north west.
Jonas Lossl and Ben Hamer have shared the Terriers' goalkeeping duties this season.