They say the cards "highlight a fundamental hypocrisy" that Christianity is seen as "fair game to mock, disparage and insult."
One of the cards, produced by Pontefract-based gift firm Love Layla, includes the wording "Mary just needs to admit she slept with someone else."
The ‘C’mon Mary’ card also features a cartoon depiction of the nativity scene with Mary, Joseph and baby Jesus.
It is part of a new range of "humorous and edgy" Christmas cards.
Others in the range include ‘Hands up if it’s your birthday’ featuring Jesus holding his arms aloft and ‘I am sorry that your December birthday is overshadowed by a bloke that wore socks with sandals’ featuring an apologetic cartoon Christ.
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But the cards have attracted a backlash from Christians on social media. Some have even called for the card to be removed from sale.
James Mildred, communication manager of Christian Action Research and Education (CARE), said: “A lot of Christians will be deeply offended by this sort of thing. It highlights a fundamental hypocrisy that Christianity is seen as fair game to mock, disparage and insult.
“As Christians we believe it is vital to exercise respect when sharing our views and listening to what others think. This card is the opposite of respectful."
Mr Mildred added: “Sadly, this happens every Christmas, which is such an important time in the Christian calendar. It is just another sad example of the kind of under the radar discrimination Christians around Europe and currently facing.”
Love Layla’s cards and gifts are no stranger to controversy. Its alternative "thank you teacher range" caused fury among teaching unions who branded the range as disrespectful to the profession.
Stacey Dennis, who founded Love Layla four years ago, started out printing her designs in her bedroom after being made redundant as a graphic designer.
Now Love Layla has turned into a multi-million-pound business, with a warehouse operation in West Yorkshire and a business in Australia.
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Stacey said: "We've really pulled out all the stops for this year’s Christmas range as this is what our customers want with some designs providing much more of a playful edge.
"And with the way this year has panned out, people are in real need of a laugh and a giggle. Christmas isn’t just about religion it’s also about having fun and enjoying ourselves.
“We’re just trying to be original and funny by catering for all different types of humour, we don’t set out to deliberately upset anyone.
"People just need to take it in the light-hearted spirit that is intended. If they want a more traditional Christmas card or something more mainstream then they shouldn’t be looking at our range.
"There’s plenty of choices available out there for those who believe Christmas and comedy don’t mix.”