This looks similar to the felt liner that came loose from my Kenmore Elite 6 year old dryer.
I called several reputable repairmen/companies and got similar quotes of $600.00 and they said it was not worth repairing, at Home Depot and Costco inquiring about replacements and adhesive was also a dead end. Googling felt liners and adhesive turned up "Seal All" (available at automotive stores and maybe Walmart). Seal All says it is not good for higher than 150 degrees F. I don’t think my dryer gets that warm, so I gave it a shot.
I tried to pry my dryer apart to get the drum out with no success. So, I put a bead of Seal All on the stitch side of the felt that adheres to the drum and shoved it back in..........WORKED!!!!!!! Clothes are drying just fine and the liner is staying in place.
My felt liner was still in great shape so I am pretty certain it would not work with an old tattered felt liner and accept no responsibility if it doesn’t work for you but it just made sense to me to try.