It is understood he has the backing of the Welsh Rugby Union to accept the position but the union is understandably keen for the negotiations to be concluded before Wales meet up for their first World Cup training week at the Vale of Glamorgan in late May.
Newly-appointed Lions chairman Jason Leonard is expected to lead the negotiations which are expected to begin within days.
A senior Lions source said: "Warren is our No 1 choice. He understands the Lions and the unique challenges of touring. He has already proven his ability to put together a successful touring squad, not just in picking form players but also the right kind of personalities and characters that make the Lions so special.
"His track record with Wales speaks for itself and the brilliant thing for the Lions is that he will have no distractions in the build-up to the South Africa tour and there will also be no complications connected to hiring him from one of the unions.
"It is important that we get the deal done as quickly as possible to allow him quite rightly to concentrate fully on Wales's World Cup preparations."
Significantly, Gatland's decision to accept the Lions post will also not necessarily rule him out of contention to succeed Eddie Jones as the next England head coach if the RFU decide to go for him.
The Telegraph revealed in February Gatland had emerged as the front-runner before he went on to win a record third Grand Slam with Wales, increasing calls from former England players and supporters for the RFU to move to recruit him.
It admitted in March it had opted to delay making its appointment to succeed Jones, who has a contract to 2021 but may step down after the World Cup in Japan depending on results.
However, it is thought that if the RFU decided to go for Gatland, his Lions appointment could be incorporated as Jones could still be in place up to the tour in July 2021 or it could afford the opportunity to give head coach experience to one of England's current coaching staff, such as Steve Borthwick, during Gatland's time with the Lions.
The RFU has been at pains to express Jones is contracted to 2021 and is also considering a mentoring role during that period – meaning Gatland could be available before the Australian has even left his position.
Nigel Melville, the RFU's interim chief executive, has yet to confirm whether or not Gatland is on their shortlist but it is thought Gatland's decision to accept the Lions job will not affect their decision.
The Lions refused to deny that they were about to open negotiations with Gatland.
A spokesperson said: "As with any thorough recruitment process it is confidential and it is therefore not appropriate to discuss until an appointment is made."