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.
His impressive form saw him touted as a potential successor to Stephen Jones at international level, with plenty of media outlets and supporters tipping him to make the summer tour to New Zealand.
Wales coach Warren Gatland decided to stick with just Jones and Dan Biggar for that two-match adventure but Tovey is expected to challenge for honours before and after next year's World Cup.
The Dragons face Cardiff Blues in the derby fixture at Cardiff City Stadium on Saturday before welcoming the Scarlets to Rodney Parade on Friday, September 24.
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