Boullier's revelation that the Woking team are looking for a "mix of experience and eagerness" suggests that Button, 34, could be the one to make way for the two-time world champion, and the Briton frustratingly admitted at Sunday's Russian Grand Prix that the wait was "actually getting funny".
However, Boullier has said that he is "comfortable" with delaying an announcement until he is certain that he has come to a decision which will set up a team that has not won a Grand Prix since 2012 to return to prominence.
He told reporters: "I always said that we are working on the strategy for the coming years - plural - which means we could take on a big name now, or even two big names for next year, or we could stay the same and go for a big name - or again two big names - for 2016.
"We are still assessing what we should do, as it is a big decision that will obviously be driving our success for the medium term. If we take the wrong decision we could be successful in the short term but maybe not in the long term.
"And we are in the comfortable position of being able to wait. I know that doesn't make it comfortable for Jenson and Kevin, but I have to think about McLaren first. We have to make sure we make the right decision."
McLaren are currently fifth in the constructors' championship with three races of the season remaining.