Martin Whitmarsh has piled the pressure on Ferrari, despite seeing his McLaren team endure a difficult Japanese Grand Prix weekend which ended with Lewis Hamilton making another Twitter gaffe.
Button and Hamilton could only finish fourth and fifth at Suzuka as McLaren struggled for pace relative to the stunning performance of the Red Bulls.
Sebastian Vettel cruised to the win to narrow Fernando Alonso's championship lead to four points.
Hamilton did at least trim 10 points off his own deficit to Alonso after the Ferrari driver failed to finish, but it was a challenging weekend all round for the Woking-based team.
The race was the first since Hamilton's move to Mercedes was confirmed and that, understandably, was the major topic of paddock discussion, even allowing for Michael Schumacher's retirement announcement.
And he completed the weekend by taking Button to task on Twitter, having been led to believe his team-mate had stopped following him on the social media site.
In a series of tweets on his official feed, which has around 1.1million followers, Hamilton said: "Just noticed jensonbutton unfollowed, thats a shame. After 3 years as teammates, I thought we respected one another but clearly he doesn't.
"Funny thing is, we are STILL teammates! All good tho, I plan on giving this team & fans all I got til I cross the finish line in brazil!!!"
But the 27-year-old was left embarrassed when he realised Button did not follow his Twitter account in the first place.
That led him to tweet: "My bad, just found out Jenson never followed me. Don't blame him! Need to be on Twitter more!"
Hamilton's comments are unlikely to have impressed Button and could put a strain on a working relationship that, while seen as being largely harmonious, has been known to be tense.
Hamilton's Twitter climbdown is not the first time he has mis-stepped on the social networking site.
At the Belgian Grand Prix he posted a picture of the telemetry from his and Button's qualifying laps, which the team quickly told him to remove due to the sensitive nature of the data, on that occasion Button said he was "disappointed".
On the track things were not a great deal better. Having had the quickest car at the previous four races, McLaren will have been concerned about how Red Bull turned the tables to dominate in Japan.
But team principal Whitmarsh believes it is Ferrari and Alonso who are the ones under the most pressure after seeing their championship lead all but wiped out.
Whitmarsh, speaking before Hamilton's Twitter outburst, said: "Let's be positive about it. I guess Fernando will be going home more unhappy than most, he is under pressure, more pressure frankly from Sebastian than from us, but we have closed that gap again and we have five races to go.
"Red Bull were on form here and at the moment Ferrari are under some pressure from us and Red Bull. If they have some big updates they may be able to change that position, but obviously I don't know."