The couple - currently pregnant with their first child - dismissed the claims, telling New magazine that they still plan to marry next year.
"Usually it doesn't bother me, but I was really upset. I couldn't sleep one night so I went on Twitter and people were saying we had split," Houghton said. "I woke Alex up and said, 'Did we split up?' It has really affected me."
Admitting that they have the "odd row", she continued: "But we're so far from splitting up - we're still getting married! But the minute we've had an argument, Alex will poke his tongue out and make me laugh. It's a bit different from packing your bags and taking your ring off, isn't it?"
The Celebrity Big Brother winner said that she had also been upset by "nasty comments" from Twitter users.
"I was really upset the other day because someone had a go because I spelt pigeon wrong. Sorry, I put a 'd' in it! But I felt like I was being physically attacked by the Twitter world. Some nasty bloke said, 'I hope you have a miscarriage', and someone else said our baby would have a low IQ. It makes me really angry and I feel protective over my child. It's disgusting."
Reid added: "It's assault. People should be careful as it can get reported to the police. Cyberbullying is disgusting."
The couple also addressed comments made by The Only Way Is Essex's Joey Essex, in which he remarked that she "used to get about a bit".
Discussing their wedding plans, the cage fighter said that it was proving "difficult" to pick a date as he is still married to Katie Price.
"We are trying [to get it sorted]," Houghton said. "We've got so much to do anyway. It will probably be next year."
The full interview appears in the new issue of New magazine, out now.
> Chantelle Houghton: 'My daughter will beat Harper Seven's fashion'
> Alex Reid: 'My daughter with Chantelle Houghton will be a bad-ass'