The midfielder - on a season-long loan arrangement at Carrow Road - hobbled out of the action at half-time against Sheffield Wednesday and was back at Spurs Lodge for treatment on Monday.
He will now undergo further scans on Tuesday to determine the extent of the injury.
"We don't think he will need an operation, so that will help a bit, but he will be out for quite a time," said Head Coach Martin Jol.
He's due for another scan and maybe it's not a 100 per cent rupture of his Achilles
We are not certain yet, so we have to wait and it's unfortunate for Dean.
"We sent him on a loan to Norwich, he's played a lot of games and that's good for his development.
"Hopefully he'll be back soon and the same goes for Mounir El Hamdaoui with his dislocated shoulder. He's had an operation and he might be out for three months, so that's a big blow for the lad as well."
Striker Mounir picked up his injury recently on loan at Derby County.
Dean Marney faces a spell on the sidelines after picking up an Achilles injury in action for Norwich City at the weekend.