Lampard doesn't need anything because the reason why Chelsea brought him here was to have a manager temporarily until making their decision on a permanent one. Therefore Lampard was never going to be their permanent manager. Pochettino and Nagelsmann have been the most rumoured names to sign for Chelsea. First they said that Nagelsmann was close to signing a contract with them but now things seem to have changed.
Well i know Lampard was basically not hired to win any trophy, but Todd Boehly also didn't sign up for this shambolic performances every week when he hired Lampard to be caretaker boss, since his return to Chelsea he has lost all five games as manager, thus i am pretty sure the Chelsea hierarchy would be confused either to sack him or allow him see out the season, but what if they do that, and with Lampard's horrific form Chelsea finishes in the bottom half of the table, closer to the relegation zone that the top four.
Having said that, both Nagelsmann and Luiz Enrique have both pulled out of the race to be Chelsea manager, thus it is just Pochetinno right now, and info from reliable sources is that the deal is very close, so Pochettino might be Chelsea's managee very soon.