The infected player, Niklas Sule, is fully vaccinated and currently asymptomatic, added the DFB.
Kimmich, who captained Germany in a recent World cup qualifier, sparked a fierce debate in Germany at the weekend when he revealed he opted not to get vaccinated, because of "personal concerns".
According to German daily Bild, Kimmich is one of five Bayern players who are unvaccinated. Kimmich says he has reservations about getting vaccinated, "especially with regard to the lack of long-term studies".
First-half goals by Thomas Mueller and Joshua Kimmich put Bayern 2-0 up before Pavard was shown a straight red card just three minutes into the second-half.
Bayern have won their last seven games, scoring at least three goals on each occasion, with Dortmund, Leipzig and Barcelona on the list of defeated opponents.
Germany cruised to a 4-0 win in Reykjavik on Wednesday to go four-points clear at the top of their group, but their plane ran into difficulties during the night flight to Frankfurt.
"We are delighted that Joshua Kimmich is staying in Munich! Joshua identifies totally with FC Bayern," the club's sporting director Oliver Kahn wrote on Twitter.
The German team, however, remains a work in progress for Joachim Löw in his last tournament as coach.
The French were flying, even if victory in the end was of the quintessential, economic Didier Deschamps blueprint.
Germany are eager to avoid a repeat of the 2018 World Cup when they finished bottom of their group and the pressure is on for Saturday's home game against Portugal.
We take a look at one player from all of the 24 teams who can make a major impact in the tournament.
Bundesliga: The importance of Lewandowski, Kimmich and others in Bayern Munich's title-winning campaign
Kimmich showed his creative side with three assists when Bayern thrashed bottom side Schalke in January.
Bundesliga: Bayern Munich on verge of ninth straight title as they beat Bayer Leverkusen to take 10-point lead
Bayern seized control at the Allianz Arena with early goals by Eric Maxim Choupo-Moting and Joshua Kimmich after second-placed RB Leipzig had earlier suffered a shock defeat at relegation-threatened Cologne.
Champions League: Bayern Munich are 'better team' and will beat Paris Saint-Germain, says Joshua Kimmich
"I believe that we will progress, because we're the better team," Kimmich told the club's website. Bayern need to overturn a 3-2 defeat from last week's first leg in Munich.
Bundesliga: Robert Lewandowski nets his 32nd league goal as Bayern Munich beat Werder Bremen; Wolfsburg thrash Schalke
Lewandowski took his season tally to 32 league goals and matched the record of scoring against 16 different teams within a season. He managed the same feat last season. Former Bayern great Gerd Müller and Bremen striker Aílton are the only others to manage it.
FIFA Club World Cup: Winning’s in our DNA, says Bayern Munich chairman Karl-Heinz Rummenigge ahead of semi-final
Bayern have already won the German league and cup, Champions League as well as the German and European Super Cups in 2020.
Kimmich damaged the meniscus in his right knee in a challenge on Borussia Dortmund forward Erling Haaland during Bayern's 3-2 win over its title rival on Saturday. He had to be helped off the field.
German Super Cup: Joshua Kimmich's late strike propels Bayern Munich to title with victory over Borussia Dortmund
Bayern have already won an eighth-straight Bundesliga title, the German Cup, Champions League and the UEFA Super Cup this year.
It was Bayern Munich’s first defeat in any competition since a 2-1 loss at Borussia Monchengladbach on 7 December, 2019
Champions League: Methodical Bayern Munich battle past Paris Saint-Germain to claim sixth European Cup
Heading into the match against PSG, it seemed rather inevitable that this new look Bayern would rip the French club from limb to limb, but the actual match turned out to be much closer than expected