The Spanish driver is already under contract beyond 2018, but that ties him only to McLaren as a brand.
Meanwhile, McLaren is expected to enter a team in Indycar in alliance with former team driver Michael Andretti's existing infrastructure.
"It's McLaren's choice, but we know they have a very good relationship with Michael," Andretti's father, F1 legend Mario Andretti, is quoted as saying by Marca.
"Fernando knows what McLaren will be in formula one in the future, and he does not have too many options in other teams. That's the problem.
"Unless he can win with McLaren next year or the year after, it's time to do something else. It's obvious that he's not ready to hang up his helmet."
Alonso is openly targeting the elusive motor racing 'triple crown', having already won the Monaco Grand Prix and now Le Mans.
The Indy 500 is his remaining target.
"Anything is possible," said McLaren executive Brown when asked about Alonso's future.
"It's clear that he wants to win the triple crown, but he's only 36 years old and still has time. We would love to keep Fernando in our formula one car."
If Alonso does go, Brown admits that Daniel Ricciardo would be a good replacement.
"We are fans of Daniel's," he said. "This is the time of year when you start to look at your driver lineup, but we don't want to say any more than that."
Alonso is seventh in the driver rankings ahead of today's French Grand Prix.