The Argentine boss's future has been a source of ongoing debate during his first campaign at Stamford Bridge.
But the Blues, who finished a lowly 12th last season, need just a point at home to Bournemouth on the final day of the campaign on Sunday to clinch sixth position.
That would mean they would qualify for either the Europa League or the UEFA Conference League depending on various permutations.
The Blues head into the weekend on the back of Wednesday's 2-1 win at Brighton, during which in-form attacking midfielder Palmer opened the scoring during Chelsea's fourth straight victory.
"For it to be in our hands to get sixth place is big for us," Palmer told Chelsea's website.
"Hopefully we can do it. Chelsea is a big club, everyone wants to finish higher than sixth but we have to look at where we've been at the start of the season, throughout the season."
The 22-year-old England international added: "We've been 13th and 12th this season. Last season was the same, so to get back into Europe is a step in the right direction.
"Big credit to the manager because all the players love him, we've got to fight for him. It's been a good season and hopefully we can get European football at the end of it."
Palmer has now scored 22 league goals this term, a tally that equals the single Premier League season best of Chelsea's all-time record scorer Frank Lampard.
"Getting 22 goals in the Premier League is a big achievement," said Palmer, who joined Chelsea from Manchester City for £40 million ($51 million) move last September.
"Obviously Lampard is one of the best Chelsea players in history, but hopefully I can keep going."
