Ledley has been a long-term target for Celtic boss Neil Lennon after seeing out his contract at Cardiff.
The Wales international attracted interest from the likes of Stoke and West Brom, but it appears Celtic are favourites to clinch his signature.
Ledley's agent, Dave Baldwin, has hinted a move is close as the midfielder has been impressed by what he has seen at Celtic.
"When Joe was here earlier, he was impressed. You couldn't fail to be impressed with a club the size of Celtic," Baldwin told the Daily Record.
Hull City midfielder Bullard spent Wednesday looking around the facilities at Parkhead and Lennoxtown as he considers a season-long loan switch.
Cash-strapped Hull are ready to loan out Bullard to get him off their wage bill and Celtic are hoping to persuade him to make the move to Glasgow.
Bullard revealed the talks had been productive, but he refused to put any deadline on when he may make a decision on his future.
"It went well," said Bullard. "But I'm not putting a timescale on anything. I'm afraid I can't really say anything more at the moment."
Celtic have already signed defensive pair Charlie Mulgrew and Cha Du-Ri this summer, while Bullard's former Hull team-mate George Boateng has been training at the club as he looks to find himself new employers.