President Trump purchased Turnberry five years after it last hosted the Open and has since spent £200m on improvements. In 2023 he publicly called for it to host the Open again.
New R&A chief executive Mark Darbon said last week he "would love" the course to host the tournament again, and that logistical issues are the stumbling block, not who owns it.
The number of spectators attending the Open has dramatically increased since 120,000 people attended Turnberry in 2009, with 278,000 spectators set to attend this year's event at Royal Portrush.
A feasibility survey has been commissioned to assess whether the venue and town is equipped to cope with such numbers, with question marks over transport and accommodation.
Darbon's stance is a significant shift from that of his predecessor, Martin Slumbers, who had strongly suggested the course would not be restored to the Open rota while Trump was associated with the venue, expressing concerns that the focus would be off the course.
That led to suggestions that Whitehall officials have pushed for the Open to return to Turnberry, with ministers keen to please Trump in the wake of his tariff policy and ongoing trade talks between the two countries.
Sources from both sides deny any pressure has been applied.
In February, it was reported that White House officials raised the possibility of Turnberry returning as a venue for the Open during the prime minster's visit to Washington. King Charles even referenced the course in a letter inviting the president to a second state visit to the UK.