Frank Lampard, who they hired on an interim basis to see out the campaign, has lost all five of his matches in charge since returning.
Chelsea had looked to be the frontrunners in the race to hire Nagelsmann, who believed talks with the west London giants were a formality.
He was, however, surprised to learn that other candidates were in the running for the post as well – and consequently dropped his interest.
That has given Tottenham an opening when it comes to making Nagelsmann their next manager, with the 35-year-old believed to be open to talks.
Certainly on Thursday night, in the second half against Manchester United, Spurs looked a bit more like their usual selves.