Palmer struggled in his debut campaign in 2016 but managed to save his drive with an improved showing in the second half of the season.
This year, though, has been more like the first half of last season with the Brit yet to get off the mark.
In sharp contrast, his teammate Nico Hulkenberg has scored 26 points to put Renault eighth in the standings.
This has led to speculation that Palmer is on the verge of being replaced, either during this season or at the end of it.
However, Abiteboul is adamant that not all is lost for the 26-year-old.
“Frankly, if he manages to turn around the situation, which he did last year, we are completely open to a future between the team and Jo for one more season,” Abiteboul told Autosport.
“Stability would be good for the team. That is also, what we wanted last year – to have Nico changing we wanted not to have to change two drivers.
“Things could go his [Palmer’s] way, but at the end of the day, that’s in his hands.
“He knows that right now he’s on a one-year contract and completely understands the team has to assess its options for the future.”
He added: “We’ve seen very good things with Jo, both during the winter tests, during the season, during a session.
“He’s capable of extracting really good pace from the car, doing a very good job, providing accurate feedback, being very committed to the team.
“At the same time there’s been a mix of mistakes, missed opportunities – clearly not assisted by circumstance with reliability that has been clearly weaker on his side of the garage.
“Very rapidly what this sort of mix has created is, I believe, a lack of confidence – a lack of confidence in himself, a lack of ability to put his head down in the difficulties that will encounter any driver in a race weekend or in a season.
“And that lack of confidence has started to kick off in a sort of snowball and has led to the situation we have now.
“I’m really trying to protect Jo and to confirm to Jo almost on a daily basis my and the team’s commitment and full support – in order to recreate the confidence in himself and in the team.
“It’s not the job of one day.”