Nagelsmann was sacked by Bayern Munich last month, but, despite wanting to take a break until the summer, he is willing to listen to what Chelsea have to say.
The Blues are impressed with his credentials and he ticks a lot of boxes, but co-owners Todd Boehly and Behdad Eghbali have concerns that he is only 35.
Nagelsmann worked at RB Leipzig with Chelsea technical director Christopher Vivell, who is a big admirer of the German.
The search for a new head coach is being led by Chelsea sporting directors Paul Winstanley and Laurence Stewart, with the final decision to be made by the owners.
There is still yet to be any contact from Chelsea over Nagelsmann with Bayern, who are due compensation if they want him.