Robins target Kevin Long looks set to stay at Turf Moor.
The Robins are said to have bid £350,000 for the Irishman, who failed to play a single minute of Burnley's 2015-16 promotion campaign.
Despite this, Sean Dyche's side will reportedly offer a new deal to the 25-year-old, denying him a reunion with Lee Johnson at Ashton Gate.
Johnson signed Long on loan for Barnsley last season, and the Sun previously reported a medical had been scheduled for the former Cork City man.
Bristol City signed Iceland international Hörður Björgvin Magnússon earlier this summer, but released Adam El-Abd and Karleigh Osborne to leave them with two senior centre-backs.
"Had we lost one, we would have had to change everything," Johnson was quoted as saying by the Brsitol Post after Saturday's win at Burton Albion.
"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 (right) and Leicester's Liam Moore have both been mooted this summer (Bristol Post), and the Robins' interest in the pair could now be stepped up.