They will miss Sunday's big league match against third-placed VfB Stuttgart and Wednesday's final game of the year against VfL Wolfsburg before the start of the winter break. The Bundesliga will resume on January 12th.
It is unclear how long they will be sidelined with both players having picked up muscle injuries.
"These are tough games. These two injuries on the right side hurt us," Tuchel told a press conference. "Both are starters and both have played really well since the start of the season. They will be out for weeks."
The Bavarians, who crashed to a 5-1 defeat by Eintracht Frankfurt for their first league loss of the campaign, are in second place on 32 points and have a game in hand.
Bayer Leverkusen, who are in top spot on 36, face Frankfurt on Sunday. Stuttgart are third, a point behind Bayern on 31.
Bayern did bounce back with a win at United in their final Champions League group match, having already secured their spot in the knockout stage of the competition.
Tuchel, however, said they still needed to show a reaction in the league for last week's loss.
"We want to win but we recognise the class of the other team," he said. "We showed against United what we had failed to show against Frankfurt last week.
"If we don't match Stuttgart's spirit and intensity then individual quality won't make the difference. There is a reaction to come (for last week's loss) and we want to show it on Sunday," he added.
