According to Bloomberg, the handsets will initially be launched in Europe in the first half of 2011, and will then be made available in the US.
INQ specialises in developing handsets with social features
An unnamed source also said AT&T, was in talks to offering the device in the US. Although no European carriers were named.
However, the social network confirmed to Bloomberg it has been working closely with INQ "to help them build a deeply integrated Facebook experience on their devices".
"While we can't speak for their future product development plans, we can say that our view is that almost all experiences would be better if they were social."
One of the handsets is thought to look simlat to the iPhone, while the second handset could feature a touchscreen and slide-out Qwerty keyboard. As well as offering access to Facebook, both handsets will also give users acess to other social media services such as Twitter.
However, despite Facebook's denial, more than two-thirds of Brits still expect the social network to launch a smartphone, says MyVoucherCodes.
Nearly a quarter (24 percent) said they would buy a Facebook mobile phone and almost a third admitted they expect the handset to outshine Apple's latest iPhone.
Of those that wouldn't buy a Facebook phone, nearly seven in ten (69 percent) said it was due to privacy. Furthermore, 43 percent were worried Facebook would have access to contacts and personal information stored on the phone.
"The Facebook brand is so strong that I have no doubt there'd be a lot of interest in a 'Facebook phone'," said Mark Pearson, chairman of MyVoucherCodes.
"Although it's clear that privacy worries would need to be dealt with before consumers would trust the information of friends and family to them."