Nagelsmann, whose sacking in March 2023 led to Tuchel's arrival, was tipped to return to Bayern, but has now extended his contract with Germany until 2026
English newspapers had linked Sanchez with a move to Pep Guardiola's side in exchange for England forward Raheem Sterling.
"No formal bid but fair to say that figure was turned down. No player involved," an Arsenal spokesperson said.
There could be an exodus of players from the Emirates before Thursday's transfer deadline with British media reporting that an agreement for Chelsea to sign Alex Oxlade-Chamberlain was close.
But Arsenal said a deal for the 24-year-old England midfielder was not agreed.
"Nothing has moved on with him yet," the spokesperson said.
($1 = 0.7741 pounds)