It is understood that Gatland was interviewed for the role of head coach, along with Ireland's Declan Kidney and Scotland's Andy Robinson, in London last week.
The 48-year-old New Zealander is the outstanding candidate for the role after guiding Wales to a second Six Nations Grand Slam in his four-year tenure and has previous Lions experience having been part of the backroom staff for the 2009 tour to South Africa.
The Lions plan to unveil their choice for head coach at the end of April or the start of May, and it is understood that initial talks have been held between Gatland, the Lions and the Welsh Rugby Union regarding the terms of his release from Wales duty for the tour.
One of the key areas of discussion will be whether Gatland takes charge of Wales for their autumn international schedule later this year.
The Lions would ideally like their head coach to be available for a 12-month period prior to the trip Down Under.
Gatland is set to head to Australia when Wales tour there in June, which could prove valuable in terms of preparation for taking the cream of British and Irish rugby back there next year, and the WRU are understood to be keen for him to also be in charge come November, before stepping aside for the 2013 Six Nations.
WRU chief executive Roger Lewis told The Guardian: "We have had informal discussions with the British and Irish Lions over the last few months and we have talked to Warren about this for the best part of a year.
"Warren does not have a clause in his contract which contains provision for him to coach the Lions, but we are hugely supportive of him and them.
"What I will say is that there is no question that Warren will not be leading Wales on our very important tour to Australia and I, personally, would like him to be involved in our autumn Tests against New Zealand and Australia in Cardiff."
Should he land the Lions job, Gatland may choose to include Wales assistants Shaun Edwards and Rob Howley, and England forwards coach Graham Rowntree, who all went to South Africa in 2009, in his coaching team.
Gatland would also become just the second overseas coach to take charge of the world's most famous touring side.
His fellow New Zealander and then Wales coach Graham Henry led the 2001 tour to Australia, where the Lions lost the series 2-1 despite an emphatic first Test win in Brisbane.
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