"We're trying to sign him, finalise the deal and get another alternative for our squad," Tuchel told reporters on Thursday.
Dier arrived in Munich on Thursday ahead of a medical check and told Sky Germany: "Bayern is a wonderful club."
Dier, whose contract with Tottenham was set to expire in the summer, arrives for a reported fee of four million euros (£3.4 million).
Often deployed as a centre-back, the versatile Dier can also play as a defensive midfielder or as a right-back.
Dier spent a decade at Tottenham and will be reunited with former Spurs teammate Harry Kane in the Bavarian capital.
Tuchel has been looking for additional support in defence, particularly with Kim Min-Jae set to miss several games as he represents South Korea at the Asian Cup.
Tuchel said he saw Dier as a "specialist in the position of central defender" and told reporters "first and foremost, this will free up Leon Goretzka for midfield".
Kim's absence means Tuchel has only two recognised centre-backs, Matthijs de Ligt and Dayot Upamecano, remaining in the squad.
Dier was born in England but grew up in Portugal, moving from boyhood club Sporting to Tottenham in 2014.
He has made 49 appearances for England, scoring three times.
Dier has only played four times for Tottenham this season.
Bayern sit second on the Bundesliga table, four points behind leaders Bayer Leverkusen, with a game in hand.
The German champions re-start their season after the winter break at home to Hoffenheim on Friday.
