Villa had an enquiry for the midfielder swiftly rebuffed by Nuremburg last month, but are now looking to finance an improved bid.
Lambert wants Kiyotake but must offload a string of high earners at Villa Park to finance an improved bid, according to The Mirror.
Club officials are reluctant to let the experienced quartet leave cheaply, with Villa holding out for transfer fees.
Once fringe players have been moved on Lambert believes that he can persuade Nuremburg to part with Kiyotake, but the Bundesliga club are believed to value the Japan international at around £10m.
Kiyotake has played 19 times for his country at senior level, including an appearance at this summer's Confederations Cup in Brazil.