After the club secured Champions League qualification in his first season as manager at his former club, they have failed to push on after a huge summer spending spree.
Lampard insisted this week that his job was not under immediate threat as he tries to find a way to get the best out of the likes of Timo Werner and Kai Havertz.
However, the ex-England midfielder will not be get forever to secure silverware, and Roman Abramovich might decide to mix things up at the end of the season unless there is a change in fortunes.
The Sun have reported that although he is in no immediate danger, Lampard could be axed as part of a summer of change if the results don’t improve over the next few months.
They have picked out Thomas Tuchel as a potential replacement for Lampard after the German was dismissed and replaced by Mauricio Pochettino at PSG.
Tuchel followed the path of Jurgen Klopp by joining Borussia Dortmund from Mainz 05, but he failed to bring them Bundesliga glory before leaving the club after a DFB Cup triumph.
The 47-year-old then led PSG to two league titles and a Champions League final, leaving the club having achieved the best win percentage in Ligue 1 history with 75.6 per cent.
Lampard said after his side lost at home to Manchester City: “I was real after we beat Leeds saying that we’re not title contenders, and I’m real now saying it takes time.
“Any build or rebuild takes pain: pain behind the scenes and pain on the pitch occasionally.
“This is a difficult period and I understand the reasons why. Today the first half showed me the reasons why and we have to keep fighting and I’m the first one who has to keep fighting.
“I’ll always feel heat. I felt heat when we were on our good run, because I know that round the corner can be a negative.
“Maybe (N’Golo) Kante you can take out, but when you look around the team, there’s a lot of youth, a lot of new players that are trying to come in and settle.”
Next up for Chelsea is an FA Cup clash against Morecambe before Fulham in the Premier League.