The Clarets are set to bid £10million for the Stoke captain, who has been with the club since 2007 after a switch from Manchester United.
The Sun says Sean Dyche has targeted the 30-year-old central defender to fill the void left by Michael Keane, who departed Turf Moor earlier this summer for a switch to Everton.
That move has left Burnley short of options at the back with only Ben Mee, James Tarkowski and Kevin Long left as recognised centre-backs.
This has led the club to try and persuade Shawcross to move back to the North West.
Burnley have money in the pot following the sale of Andre Gray to Watford on Wednesday for a fee of £18.5 million.
Stoke may be willing to let Shawcross, who has only a year left on his contract, go following the arrival of Kurt Zouma on loan.