The 21-year-old, who came through the ranks at Turf Moor, spent last season on loan to Morecambe, where he made 43 appearances for the Shrimps in all competitions.
Bury manager Lee Clark told the official club website: "We felt that we needed another centre-half in. We didn't want to leave ourselves short and we always want to have at least one central defender as a replacement on the bench.
"I have seen Alex progress year on year at different levels. I have seen him in the National League for Gateshead and then last year for Morecambe.
"We feel he has got that progression in him. Whenever I've dealt with Burnley as a club and I've signed their young players they've all had a great desire and attitude to do well and we believe Alex ticks those boxes."
Whitmore's spell with Bury will be his fourth loan in total and his second in the EFL.