Kevin Long could swap Turf Moor for Ashton Gate.
City did not have a centre-back on the bench during Saturday's win at Burton Albion, and manager Lee Johnson told the Brsitol Post afterwards:
"Had we lost one (a centre-back), we would have had to change everything.
Bristol City manager Johnson wants another centre-back
"It's not easy to get a quality player who is ready to go with the money currently flying around in the Championship, but in an ideal world, we will get another centre half in."
Moves for Barnsley's Alfie Mawson and Leicester's Liam Moore have been mooted this summer (Bristol Post) but Long appears to fit the bill for now, according to the Sun.
Leicester's Moore is a reported Robins target
The report claims Burnley are ready to sell the Irishman, who failed to feature in last season's promotion campaign, for a fee of around £500,000.
Like Mawson, Long played under Johnson at Oakwell last term, before ending the campaign on loan to Milton Keynes Dons.
Burnley's Long in action against West Ham
The 25-year-old arrived at Burnley at the start of the decade from Cork City, but has played just 27 times for the Clarets and is not said to feature in Sean Dyche's plans this season.
Bristol City, who sit second in the Championship on goal difference, added Icelandic centre-back Hörður Björgvin Magnússon to their ranks this summer, but released Adam El-Abd and Karleigh Osborne.
Magnusson (c) has partnered Aden Flint so far