The 22-year-old Bath man will have a scan on Saturday.
If Watson, who missed the autumn internationals with a broken jaw, is ruled out, then it could present an opportunity to Gloucester's Jonny May.
England, Grand Slam winners in 2016, start their title defence at Twickenham on 4 February.

The presidency has published a "transparency charter" defining her duties, funded from Mr Macron's budget.
She will have an active role in civil society, focusing on disabled people, education, children, women's issues, health and gender equality.
A petition signed by more than 316,000 people opposed a "first lady" role.
Much French media comment has focused on the 24-year age gap between her and her younger husband. She played a prominent role in his election campaign and is seen as a strong influence on him.
The social issues that will particularly engage her also figured as priorities in Mr Macron's manifesto.
The ElysÃ©e Palace's new "charter" (in French) is an innovation - no previous French presidency has stated the spouse's role so clearly.
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Creating an official first lady position would have required a change to the law or constitution. Instead, the charter uses the term "the head of state's spouse".
It stipulates that Mrs Macron will not be paid a salary for exercising her duties, nor will she have her own budget or expenses claims. The costs will come out of the presidential budget.
She will have the assistance of two advisers, her own office and a secretariat.
In his presidential campaign, Mr Macron stressed that politicians should not employ relatives, after his conservative rival FranÃ§ois Fillon lost much support for doing so.
In an interview, Elle magazine asked Mrs Macron how she had reacted to hostile and sexist comments on the age gap between her and Mr Macron. She answered: "Badly - but finally I told myself: 'Well, you're having a hard time, but just keep quiet about it.'" She added: "In time, it goes away".
She admitted that her love for Mr Macron had caused some private anguish.
"The children often suffer. I know I caused some harm to my children, and that's the thing I reproach myself for most of all. But I couldn't help doing what I did. There are moments in life when you have to make vital choices."
On the age gap, she said: "So regarding the 20-year gap, whatever could be said about that was so petty. Of course at breakfast I sit there with my wrinkles, he with his freshness. But that's just the way it is."

Nico Ramsay, 19, was found with a stab wound to the chest in Napier Avenue, Southend, on Saturday 13 February. He died in hospital four days later.
Essex Police said officers are looking for Allcie Houlder, 21, in connection with the death.
He was last known to be living in Mantle Way, Stratford.
Mr Houlder has links to Southend, Romford and Newham.
He is described as 5ft 11in (1.8m) tall, with dark skin, and has a small scar on his chin and the word Tarnia tattooed on the right side of his neck.
Mr Houlder is believed to have left the UK and taken a ferry from Dover to France on 19 February.
Mr Ramsay's family issued a statement paying tribute to him.
"Nico was a lovely, good-natured, charismatic and humorous young man who had only just turned 19," they said.
"He had a life full of hope and promise ahead of him."
England winger Anthony Watson is a doubt for the Six Nations opener against France after suffering a hamstring strain.
[NEXT_CONCEPT]
Brigitte Macron, the wife of French President Emmanuel Macron, is getting an official role but without the title "first lady".
[NEXT_CONCEPT]
Murder inquiry detectives have named a man they want to speak to in connection with the death of a teenager who was stabbed in Essex.