The Briton's German team mate Nico Hulkenberg has taken all Renault's 18 points from seven races while Palmer, in his second season with the team, has struggled.
The 26-year-old, who scored only one point in a difficult 2016 debut year, has finished 11th in his last two outings, however.
"No one is safe in F1," Renault Sport F1 managing director Cyril Abiteboul told the motorsport.com website when asked if Palmer was safe from the threat of being replaced and whether the team had an alternative.
"F1 is not an environment where anyone can say loudly, 'I'm safe'. To a certain degree there are two questions. There is Jo, and there is a second driver. Right now my focus is on Jo. Jo has to deliver.
"The fact is that Jo has a car which is a point-scoring car, and he has to enter into the points. Full stop."
Renault are currently seventh in the championship and Abiteboul said he hoped to go into the August break, after four more races, having moved up to sixth.
The Frenchman set a pre-season target of finishing the year in the top five with possible podium finishes.
Hulkenberg has scored points in four races so far since moving to Renault from Force India at the end of last year with a best of sixth in Spain. He was eighth in Canada last weekend.
"We see that there is a very substantial gap between Nico and Jo in qualifying, and in the race," said Abiteboul. "We are discussing on a regular basis obviously with Jo to see how he can improve."
Renault tested their ex-driver Robert Kubica in Spain, with the Pole, who has not raced in Formula One since he partially severed his forearm in a 2011 rally crash, making a good impression and doing 115 laps.
They also have Russian Sergey Sirotkin as a development driver.
(Reporting by Alan Baldwin; Editing by Alison Williams)
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