Tovey played just nine minutes of the victory over Glasgow at the weekend before limping off to be replaced by Matthew Jones and an X-ray has shown the full extent of the injury.
The 21-year-old was at outside-half for the Dragons on Sunday after featuring at full back in the defeat to Connacht on the opening weekend and his absence will give summer signing Jones a run in the No10 jersey.
"It's a real blow for us," said Dragons rugby manager Robert Beale.
"We will be treating him as often as possible to try to get him fit again as soon as we can."
Tovey was one of the region's standout performers last term, ousting New Zealander James Arlidge from the first XV as he made a hugely promising move from full back to fly-half.
Although he missed out on the summer tour to New Zealand, Tovey is being tipped to challenge for international honours before and after next year's World Cup.
Meanwhile, Cheeseman is hoping to establish himself in the country of his birth having spent six seasons in England for whom Bath is his only professional club to date.
The Swansea-born 24-year-old previously starred for Wales U19 and is now delighted to be back in Wales.
"I'm really happy to be coming back to Wales. I played here for many years before moving to Bath and it's great that the Dragons are giving me the opportunity to come back," said Cheeseman.
"I have to admit it's a bit nerve-wracking to be coming into such a big weekend with the away game at the Blues.
"It's hard work learning all the moves in a short time but I've already done a training session and the players are really easy-going. The coaches have been very helpful and I'm really looking forward to pulling on a Dragons jersey."