The 23-year-old and agent David Baldwin visited Celtic Park for discussions with club officials on their latest visit to Scotland.
The midfielder looks set to commit to Celtic on a long-term contract after being offered a reported £25,000 weekly wage.
Celtic also remain hopeful of signing Hull midfielder Jimmy Bullard, who remained in Scotland on Thursday after visiting the club's Lennoxtown training complex.
Championship club Cardiff would expect several million pounds of compensation should Ledley move to an English team but FIFA's transfer age limit is a year less for cross-border rules, meaning Celtic would acquire the player for free.
A host of Premier League clubs have been linked with interest in Ledley, including Stoke, Liverpool, West Brom and Blackburn.
Ledley has made more than 250 appearances for Cardiff after making his debut aged 17 and has accumulated 32 caps for his country.
Hull midfielder Bullard spent on Thursday afternoon on the banks of Loch Lomond spectating at the Scottish Open golf tournament after his latest talks with Celtic.
Hull are keen to release the 31-year-old from their wage bill following their relegation to the Championship and could sanction a loan deal which would see them pay a portion of his salary.
Lennon has so far signed two full-backs this summer in South Korea international Cha Du-Ri and former Aberdeen player Charlie Mulgrew.
He is also awaiting an answer from former England defender Sol Campbell, who finished last season with Arsenal, after holding talks with the veteran last month.
Former Hull and Middlesbrough midfielder George Boateng, 34, trained with Celtic this week, although the likelihood of him moving to Glasgow may have diminished after becoming the victim of a reported attempted sabotage of his car wheels.
Lennon and his coaching staff have also been assessing 22-year-old Montenegro forward Slaven Stjepanovic and 25-year-old Le Mans full-back Ibrahima Camara in training.