The Irish side announced the departure of Williams from his post earlier this evening. Williams is a former coach of New South Wales Waratahs.
The appointment is set to be announced at a press conference in Durban, South Africa tomorrow afternoon.
McGeechan announced last year that he would be standing down after the World Cup this autumn to take over as Scotland's director of rugby.
Leinster chief executive Mick Dawson said: "We are disappointed to lose the services of Matt, yet we do understand the attraction of moving on to an international position."