Under LPGA Tour rules England's best-known woman golfer has to win a fifth major or two more of their regular events to earn a place.
On the eve of the Ricoh Women's British Open Davies, a member of every Solheim Cup side since its launch in 1990, said: "I've had about eight goes of winning this tournament to get in the Hall of Fame.
"Who knows? This year might be the one - it would be the perfect place to do it.
"I've been playing well all year. Putting has been the biggest problem, but I putted nicely last week," added Davies, who finished joint runner-up in Austria.
"It would be nice if one of the home girls could win it and hopefully I'll be in the mix - it would just end such a great summer of sport for Great Britain."
On such a tough links and with the likelihood of strong winds Davies, winner at Birkdale in 1986 before it became a major, issues a warning not to expect the greatest spectacle of golf ever seen, however.
"We all know how good we are and we want to show that, but unfortunately this week I think you're going to find bad scoring.
"We're probably not going to look like we are that good, purely because of the conditions. I've heard people talking about eight over winning - that might be a bit extreme, but they might not be that far off."
Scotland's 20-year-old Carly Booth is the leading money-winner in Europe this season, but this is her first major - 99 fewer than Davies.
Booth has lots of local support as her father Wally, a Commonwealth Games wrestling silver medallist, used to work as a minder for The Beatles in Liverpool.
"He's told me so many stories. They asked him to go to America, but he said no because he was in training," she said.
There is also a connection to Andy Murray. Aged 10 Booth competed in the Dunblane junior championship against Murray's brother Jamie and despite being six years younger lost only on the 17th green.
England's Melissa Reid is another of the home hopes. It is only four months since her mother was killed in a car crash, yet just a few weeks later she won a tournament in Prague.
"My entire life went a complete 180," the 24-year-old from Derby said. "Some days have been harder than others and I probably have a lot more on my mind that I would normally when I play."