The Cottagers were beaten 2-0 at relegation rivals Southampton on Wednesday night.
The fourth straight Premier League defeat leaves the west London club in 19th place, 10 points from safety with 10 games left.
Reports on Thursday suggested the Italian, appointed in November, was set for showdown talks with the club’s hierarchy – CEO Alistair Mackintosh and vice-chairman Tony Khan – after which he was expected to lose his job.
When contacted by Press Association Sport on Thursday afternoon, Fulham said the manager had taken training as planned, but offered no further comment.
Ranieri, who took Leicester to the Premier League title in 2016, has overseen only three victories, with 10 defeats from 16 games in charge.
Some sections of the Fulham fans have already lost faith in the 67-year-old, chanting “you don’t know what you’re doing” during the 2-0 loss at Southampton.
A banner containing the words: ‘Risk free? Are you watching Tony Khan?’ was unfurled in reference to comments made by the club’s vice-chairman following the Italian’s appointment in November.
Speaking after the match, Ranieri maintained he did not know if his position was under threat. Asked about his future, the Fulham boss replied: “My job? I don’t know. Ask the owner.”
Ranieri, who suggested he may speak to Fulham owner Shahid Khan on Thursday, remained defiant over the team’s survival hopes.
“Something could happen. If you don’t believe, it’s done,” he added. “You have to believe. You have to fight. If you don’t fight, already you’ve lost.
“Always in my career, I’m used to working hard. I know it’s very difficult but until mathematically it’s impossible we continue to fight.”
First-team coach Scott Parker, who used to play for the club, has been suggested as a possible interim appointment should Ranieri be sacked.
Slavisa Jokanovic had guided Fulham into the Premier League with victory over Aston Villa in the play-off final at Wembley, but was dismissed after a poor start to life in the top flight.