Livingstone has hitherto refused to underwrite the games, owing to fears that doing so would leave the GLA out of pocket.
Smith has told championship organisers that if the GLA cannot be persuaded be sign the contract, other arrangements will be made.
A source close to the scheme said Smith would have to decide if the stadium was worth building if the championships failed to win backing.
He said: "It would be hard to justify it without the games." The initial contract to hold the games in London, usually signed by the host city, was drawn up before the GLA was founded, and was signed by governing body UK Sport. The news came as UK Athletics revealed the designs for the proposed £95m National Athletics Centre at Lee Valley on Thursday. The 43,000-seat stadium, warm-up track and associated facilities will be designed by FaulknerBrowns. The team includes TTH Stadium Consultants, structural engineer Anthony Hunt Associates and BDP Environmental Services.
Secretary for the event organiser Bill Glad said it was "completely confident" a solution would be found.
The Department of Culture, Media and Sport declined to comment.
