Only finer details have still to be finalised for the 24-year-old to move to Tynecastle Park. If those are ironed out then he could travel north over the weekend to complete the switch.
A season-long deal has been discussed and City are happy to loan Moore out as he is currently out of favour at Ashton Gate. Several other clubs in England are also interested in his services.
Hearts want to ensure they get his signature before Tuesday night’s transfer deadline and will continue talks in an attempt to tie everything up quickly.
Moore’s contract at Bristol City has another two years to run but he has managed only one appearance so far this season, which came in an EFL Cup tie at Forest Green Rovers earlier this month.
He has become Hearts’ main target as they try to reinforce their defence. The Edinburgh club did enquire about the former Rangers defender David Bates, currently in Germany with Hamburg, but Moore is the preferred option.
Liverpool forward Ben Woodburn completed his loan move to Scotland earlier this week and is expected to debut for Hearts against Dundee United on Saturday.
He joins Beni Baningime, Josh Ginnelly, Ross Stewart, and Alex Cochrane as summer recruits, with Cameron Devlin due to arrive soon from Australian club Newcastle Jets.
Devlin’s work permit was approved earlier this week and he has verbally agreed a long-term contract with Tynecastle officials.