The Belgian, on loan from Manchester United, picked up the injury during Sunderland's 3-2 defeat by Crystal Palace on Saturday.
A scan on Tuesday confirmed the 21-year-old has suffered ankle ligament damage.
Januzaj has scored once for Sunderland in seven appearances and was sent off against Tottenham earlier this month.

A fire crew had been called to an out-of-control bonfire with gas canisters in it, at a park opposite Southview on Thursday night.
Seconds later, two youths were hurt when the fire flared up in their faces.
The crew from Larbert gave first aid until paramedics arrived. One youth was taken to hospital for treatment.
Watch Manager Graham Cuthil said: "It's hard when your crew are being threatened and abused for trying to do their job and keep people safe.
"This bonfire was clearly unsafe which was sadly proven as the two young lads received burns to their faces before we could get to work and extinguish it."
Overall, the Scottish Fire and Rescue Service said it was a successful Bonfire Night with officers attending 144 fires across the east of the country.
Gordon Pryde, group manager for the Stirling and Clackmannanshire area. said: "On the whole Bonfire Night 2015 passed without significant incident.
"On 5 November we received 10 calls to bonfires in the Stirling and Clackmannanshire area.
"Over the weeks and months leading up to Bonfire Night we work hard with our partners and communities and we are beginning to see the positive impact of that work. We would like to thank the public for their fantastic support."
Officers in Fife and Edinburgh said they were also pleased with the relatively low level of incidents, and said it showed that education and other preventative work was paying off.

Cesare Prandelli resigned as Azzurri coach after they were knocked out in the World Cup group stage.
Former Italy midfielder Conte, 45, quit as Juventus boss last month, having won the Serie A title in each of his three seasons in charge.
He has also managed Bari and Siena to promotion from Serie B, as well as having spells at Arezzo and Atalanta.
The Italian Football Federation said in addition to his salary, Conte would receive a bonus for qualifying for Euro 2016, an added bonus for climbing at least five places in the Fifa rankings - Italy are currently 14th - and a third bonus if they reach the final of Euro 2016.
Conte is the second big appointment by the Italian Football Federation this week.
Carlo Tavecchio, who made a racist comment during his election campaign, was named as its new president.

14 January 2016 Last updated at 07:55 GMT
A special ship will drill more than 5000 metres down through the ocean floor to reach the super hot rocks of the planet's crust.
The team have picked an area in the Indian Ocean because the earth's crust is thinner there than in other places.
They also hope to find out if anything can survive the heat and pressure to live down there.
Watch Leah's report to find out more...

They arrested 57 people in Barcelona, Valencia, San Sebastian and Las Palmas in the Canary Islands.
Chinese nationals illegally living in Spain reportedly paid the ring €8,000 ($8,960; £6,820) each to obtain fake contracts as domestic workers.
In return they were given residency permits to work in Barcelona.
"They would return to Barcelona once they obtained a residency permit to work in the textile sector, without ever carrying out the job they were hired for," a Spanish police statement said.
The ringleaders, a Chinese couple,  allegedly paid employers between €1,500 and €2,000 for the contracts.
Most of the employers allegedly involved are linked to the hotel trade in Lanzarote, one of the Canary Islands.
Chinese immigrants are the fifth biggest foreign community in Spain with almost 170,000 people as of 2015, according to the National Statistics Institute.

The skull of an adult male was found by the M54 slip road on Thursday by workers surveying a roundabout near Telford.
The find sparked a West Mercia Police inquiry and led to the discovery of "further skeletal remains".
The remains have now been recovered and post-mortem tests on Tuesday will help to identify him, police said.
Officers said the remains had been there for at least two years.
Police have contacted neighbouring forces asking for information about people who had been reported missing.
Sunderland winger Adnan Januzaj has been ruled out for a minimum of six weeks with an ankle injury.
[NEXT_CONCEPT]
Firefighters were threatened by youths, moments before a dangerous bonfire left two young people injured in Stenhousemuir.
[NEXT_CONCEPT]
Italy have appointed former Juventus boss Antonio Conte as their new coach until 2016.
[NEXT_CONCEPT]
Scientists hope to find out more about how our planet was formed by studying the rocks buried deep under our feet.
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Spanish police say they have dismantled a nationwide network that illegally regularised the working status of Chinese immigrants in Spain.
[NEXT_CONCEPT]
A post-mortem examination is to take place on human remains found near a motorway slip road in Shropshire.