Newport say the decision was made at a meeting with the club's chairman Jenkins after Monday's goalless draw at home to Walsall.
With two games remaining Newport are 20th in League Two on 49 points, eight above the relegation zone.
"I can't thank Nelson enough for his hard work, commitment and loyalty, especially as he left his young family in Portugal to take on the role," added Jenkins, who completed a takeover of the club in January 2024.
"He took on a tough challenge with lots of changes at the club from a squad overhaul to the way we operate and work.
"His main goal was to keep the club in League Two during a season of transition so that we can start building for the future. It took us longer than we probably wanted, but he feels that with the job completed then it was time for him to look for a new challenge.
"Having taken on board his thoughts and the timing, it also allows us the space to reflect and plan over the final two games and the summer to appoint a replacement as quickly as possible."
Jardim took charge of the club following the departure of Graham Coughlan in June 2024 with Jenkins promising a "new direction" for the club.
The Exiles then signed 17 new players during last summer's transfer window, while Jenkins is now looking to make his second managerial appointment in the coming months.
"It's been a difficult season at times, but considering all the experience we have gained over the last 12 months then I believe there is a clear understanding of what we need to move us forward as a team and a club," said Jenkins.
"We will be looking to bring in the right manager or head coach to build on those last 12 months so we can start the season strongly; move the club forward and challenge for the top part of League Two."
The club say they will update supporters on the management team to lead Saturdays' fixture at Fleetwood "in due course".