Chief executive Mark Darbon says the R&A are having “ongoing dialogue” with Muirfield with a view to returning The Open Championship to the Gullane course for the first time since 2013.
Darbon also says there are “challenges” with bringing the event back to Turnberry.
Royal Lytham & St Annes will host The Open in 2028, with this year’s 154th tournament taking place at Royal Birkdale and next year’s staging at The Old Course at St Andrews.
Muirfield, a 16-times Open venue, will host this year’s Women’s Amateur Championship from 22-27 June.
“We’ve got a great ongoing dialogue with Muirfield,” Darbon said.
“There are some things we need to overcome to facilitate a modern Open Championship – the practice ground is one of those, along with a few on-course infrastructural points.”
On US President Donald Trump’s Turnberry, which hosted the last of its four Opens in 2009, Darbon commented: “We really like the golf course.
“But there are challenges around staging a modern Open – primarily road, rail and accommodation infrastructure.”
The Scottish Open has signed a deal to stay at The Renaissance Club in East Lothian until 2030, with the tournament played in the week before The Open to allow players time to adapt to seaside links courses.