But Neil Maxwell, who fielded a renewed appeal from one of the Counties on Thursday, said any Lee stint overseas would "most definitely" have to be made with the approval of Cricket Australia (CA).
Australian cricket's governing body has opposed the move because it wants players to be fresh before the Ashes series against England this summer ahead of a demanding year of national commitments.
"We would listen to them (CA) very closely, they're the ones who have to plan for Brett in the future, so it would be a joint decision," Maxwell said on Thursday.
Maxwell said Lee would not be available at the start of the English season next month.
"From our perspective it won't be straight after the New Zealand tour because Brett's got some holidays booked, but there may be some merit to play in some County games before the tour starts," he said.
"There's no deadline as far as we're concerned because they're keen to get him, even if it was for one game, from a cricket and a marketing point of view."
Maxwell said the idea of Lee playing for an English County had nothing to do with regaining fitness, as he was "extremely fit" despite his limited match time, and instead was more about him getting regular matches.
Attempts to get Lee to play for New South Wales (NSW) in the Sheffield Shield final against Queensland getting underway in Brisbane on Friday appear doomed.
Lee, who will again be Australia's 12th man for Friday's second Wellington Test against New Zealand, was even offered a charter flight to Brisbane to make a late start for New South Wales.
Lee said it only failed to happen because the NSW team felt the unprecedented Cricket Australia-approved move was too risky due to him missing the first two sessions of play at the Gabba.