The 25-year-old centre, who was named HSBC Player of the Tour when the Lions travelled to South Africa in 2009, sustained a knee injury in Saturday's win over England.
He suffered a reoccurrence of a medial ligament problem just five minutes into the 19-12 success at Twickenham.
Roberts played on for the remainder of the half but had to be replaced by eventual match winner Scott Williams at the start of the second period.
"We are hopeful that Jamie will be fit for our final game against France (on March 17)," said Wales backs coach Rob Howley.
"He was clearly not right after taking a knock and he gave away two penalties for offside, which is not like him."
Roberts' absence will be a big blow for Wales despite the impressive performance from Williams as the Cardiff Blues star is beginning to recreate the kind of form he showed for Britain and Ireland's elite three summers ago.
Having seen his first Lions tour ended after a heartbreaking 28-25 defeat in the second Test due to a wrist injury, Roberts struggled to hit the heights of 2009 on his return to the UK.
But the 2011 World Cup saw Roberts produce some scintillating displays for his country as they reached the semi-finals for the first time in 24 years.
His involvement in the early stages of this year's 6 Nations was in doubt until the last minute as he spent six weeks on the sidelines with knee problems but he returned in time to play a key role in a thrilling 23-21 win over Ireland in Dublin.