The 24-year-old, capped 32 times by England, has been out of action since November with a knee injury.
However, it is understood his recovery has progressed quicker than expected.
Vunipola could now be in contention to face Scotland on 11 March, or Ireland a week later.
He was one of England's outstanding performers throughout their unbeaten year of 2016, and his ball-carrying and dynamism has been missed by Eddie Jones' side.
There were initial fears Vunipola would miss the whole of England's Six Nations title defence, but it is understood he has looked sharp on his return to full training this week, and is targeting a quick return to international rugby.
Vunipola's older brother, Mako, recently made his own comeback from a knee problem, and came off the bench in the win over Italy.

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The 17-year-old from Gloucestershire won the champion apprentice trophy in his maiden season this year.
Marquand has racked up 68 wins across the last 12 months, prompting his trainer Hannon to make the comparison with Moore.
"I think he could be a champion jockey one day," Hannon told BBC Points West.
"I would be amazed if he does not go right to the top, he is the name on everybody's lips at the moment."
Marquand's rapid rise has seen him nominated for the BBC Young Sports Personality of the Year 2015 award.
The jockey has been named on the final shortlist for the award, alongside gymnast Ellie Downie and para-swimmer Tully Kearney.
"What he has achieved is a little bit special," added Hannon. "He is a lovely guy and thoroughly deserves his success. He is very popular and he has endeared himself with everybody."
Hannon, who has 300 horses based at his stables near Marlborough, is the son of former flat jockey champion Richard Hannon Sr.
Moore has enjoyed great success under the guidance of Hannon and won the champion apprentice title in 2003.
Since then, Moore has established himself as one of the most respected flat jockeys in horse racing and was champion jockey in 2006, 2008 and 2009.
"Ryan Moore performs on the big stage and he is the best jockey in the world at the moment. He is so professional and the ultimate jockey and that's who you want to be," said Marquand, who will find out if he has won the BBC award on Sunday.
"It is a yard that is constantly in the limelight and it is big to be attached to somewhere like this."
England number eight Billy Vunipola is set for a shock early return to action for his club Saracens against Newcastle on Sunday, boosting his hopes of a 2017 Six Nations call-up.
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Teenager Tom Marquand has been tipped to follow in the footsteps of three-times champion jockey Ryan Moore by his trainer Richard Hannon.