Tuchel played down the bad blood between him and Conte during and immediately after the game.
Former Chelsea and Inter Milan boss Conte replaces Nuno, who was fired on Monday after 17 matches in charge, with Saturday's 3-0 home defeat by Manchester United in the league his final game with Spurs.
Vidal made 96 appearances for Barca in two seasons, scoring 11 goals and providing 10 assists in all competitions.
Lautaro Martinez played chief tormentor as Inter Milan claimed an emphatic 5-0 victory against Shakhtar Donetsk to reach the Europa League final.
Inter posted a photograph of the Belgian forward wearing the shirt on the club's Instagram account one day after confirming that they had signed him from Manchester United.
Inter will be hoping that Conte will have the same effect on them as he did on Juventus.