An organisation based in Boston is calling on the NTS to acquire the South Ayrshire island to add to the charity's portfolio.
And in order to facilitate the bid, the backers have reportedly offered to help bankroll the move.
James Hare, executive director of the Boston-based foundation, said: "A donor realised the island is up for sale and believes it would make sense for the NTS to acquire it.
"There is a small group of our donors who are involved. There are serious negotiations taking place.
"Ailsa Craig is a world-famous island and it overlooks the jewel in the crown of the trust's portfolio, Culzean Castle."
The island is home to the largest gannet colony in Europe, a 500-year-old castle and a lighthouse. Most of the world's curling stones come from its granite quarry.
A spokeswoman for the NTS said: "It's true there is an appetite from a US donor the trust considers taking on Ailsa Craig.
"We have a rigorous process to go through before committing."
The 220-acre island has been on the market since 2011.The original asking price was set at £2.5m but was recently reduced.
Ailsa Craig is owned by the Scottish peer Archibald Angus Charles Kennedy, who is the 8th Marquess of Ailsa.
Indian tycoon Bobby Sandhu also retains an interest, having bought up former lighthouse properties 10 years ago for £85,000.