The McFly singer and his professional dance partner are self-isolating separately for 10 days after catching COVID-19 one day after they took part in the first live show of the series alongside all the other contestants.
Fletcher commented on Dowden's Instagram post announcing her positive test result: "Rest up partner. We’ve got to pull something special out of the bag for week 3."
The pair will miss the first dance-off of Strictly 2021 and be allowed to return to compete in Week 3.
The 31-year-old Welsh dancer - who came second in 2019 dancing with CBBC presenter Karim Zeroual - told her supporters: "Unfortunately Tom and I have tested positive for COVID-19, I'm not suffering any symptoms but am now isolating at home.
"I have no doubt that over the next 10 days my mind will be going crazy with choreography ideas ready to get back on the strictly dance floor for the live show in week 3!
"Thank you for all of the support and good luck to all of the couples this week. I will be cheering you all on!"
A BBC spokesperson said in a statement: "Tom Fletcher and Amy Dowden have tested positive for COVID-19. The pair are now self-isolating separately following the latest government guidelines. While they will both miss Saturday’s live show, Strictly Come Dancing protocols mean that all being well, they will return the following week.”
Watch: McFly say Tom Fletcher has hidden dancing skills
The pair performed the very first dance of the series on Saturday night, doing the Cha Cha to September by Earth, Wind & Fire.
New judge Anton Du Beke tipped them to make it through to the final, telling Fletcher: "A fabulous start - well done you, you've set the bar. Straighten up your leg actions and I think that will see you all the way through my old sausage."
Revel Horwood said: "I thought it was a little bit wooden, but what you do have is amazing rhythm.
And Motsi Mabuse said: "I think we are going to see you go far. With that start, what can go wrong?"
The All About You singer has previously admitted feeling under pressure given he's following in the footsteps of bandmate Harry Judd, who won the show in 2010.
Big Brother's Little Bit On The Side presenter AJ Odudu became the one to watch this weekend, after scoring an incredible 34 for her Jive. Judge Shirley Ballas gave her a score of 9 for the first time ever for a dance in week one.
They were both emotional after dancing the Tango to Blue Monday by New Order, to a standing ovation from some of the judges
Watch: Who's judging this year's Strictly?