After Twitter user Heather Yemm questioned KLM about their breastfeeding policy, the company replied that while it was "permitted", mothers may be asked to cover themselves "should other passengers be offended by this".
It later clarified: "As an international airline company, we transport passengers with a variety of backgrounds. Not all passengers feel comfortable with breastfeeding in their vicinity and sometimes these passengers complain to the cabin staff."
Twitter users quickly criticised the airline's approach.
One said: "May I respectfully suggest you eat with a blanket over YOUR head?"
Another added: "I'm a vegetarian and animal rights supporter. I find the thought/sight of other people stuffing dead animal flesh into their mouths incredibly offensive. Please can you confirm if you'll ask my fellow passengers to cover their heads and food whilst they eat?"
Others said they would not fly with KLM again, or would deliberately flout the policy.
The row follows a post on Facebook by a mother who claimed she was told to cover up on a KLM flight from San Francisco to Amsterdam.
Shelby Angel, from Sacramento, California, claimed she was told by a flight attendant: "If you want to continue doing the breastfeeding, you need to cover yourself."
She claimed: "I told her no, my daughter doesn't like to be covered up. That would upset her almost as much as not breastfeeding her at all. She then warned me that if anyone complained, it would be my issue to deal with."
In response, KLM said it understood the passenger's disappointment and would "relay your message to our colleagues".
Other airlines have weighed in on the Twitter thread with their own policies.
EasyJet said: "We support breastfeeding mothers and you can feed your baby on board at any time."
British Airways said it had no official policy on the issue.