It appears the green-coloured team has lost the support of its founder Tony Fernandes, who closed down his Twitter account late last week with the alarming news that "F1 hasn't worked".
Kobayashi returned to F1 this year to lead Caterham's charge from the cockpit, so with the mere survival of the team now in doubt, the Japanese claimed he is trying to look away from the speculation.
"I'm not really looking at that," he told Britain's Sky.
On the other hand, the 27-year-old can hardly ignore it.
"Of course, I'm not wishing these stories. I'm here for driving -- me, I can't change anything, you know? I cannot bring any money or stuff."
That, it seems, is precisely the problem. Malaysian aviation entrepreneur Fernandes warned over the winter that Caterham was being given just one more season to shine.
Eight races in, traditional back-of-the-grid rival Marussia has finally broken through, stranding Caterham with no points and the prospect of no more free money.
"You can understand why he's sort of fallen out of love with F1 because they haven't scored a point in five years," former team driver Karun Chandhok said at the Goodwood Festival of Speed at the weekend.
"Tony's spent an awful lot of money along with his business partner Kamarudin. (But Marussia) have scored two world championship points and it's highly unlikely you are going to beat them.
"So at the end of the year it could mean he has to spend another $30 million. I could think of a fairly big reason to stop loving formula one," Chandhok added.
"He's got a football team that's back in the premiership, a motorbike team that is doing exceedingly well in Moto2 -- and it costs him a tenth of what it costs to run a formula one team.
"Is it (F1) giving him the return he wants? Probably not," said Chandhok.
The Indian, however, said that even despite Fernandes' ominous tweet, reports of impending sale or closure for Caterham are for now "just rumours".
But team driver Kobayashi said Caterham's money problems are not just speculative.
"Financially, I think we're in a very difficult situation but I cannot do anything so I just focus on my job," he said.
"I just hope that somebody can help this team. This is the only thing I can say."