Prasad played a key role in Sri Lanka's first Test series win in England when, in the second and final match of the 2014 campaign, he took five for 50 in the second innings at Headingley, the headquarters of county champions Yorkshire in Leeds, northern England.
His haul included all of England's top four, in a match Sri Lanka won by 100 runs after clinching victory with just one ball to spare on the final day.
However, this trip saw the 32-year-old injure his shoulder during Sri Lanka's first tour match against Essex last week.
"We're hoping that we can get him bowling again pretty soon," Sri Lanka coach Graham Ford told BBC Sport after the tourists' final warm-up match, against Leicestershire ended in a draw on Sunday.
"Prasad has had a few tests and the specialist came to see him today," the South African added.
"We'll have to build him up and get him bowling a number of overs in the nets to have him prepared potentially for the second or third Test."
Sri Lanka's top-order have struggled in England so far this tour but Dimuth Karunaratne made a hundred and Lahiru Thirimanne hit 40 at Grace Road.
"As we have gone along players have started to find their feet and find a bit of form," Ford said.
"Every day, there is someone new who is starting to look like he feeling a bit more settled in English conditions.
"Hopefully with a bit more work the boys will have some good belief about what they can do. It is going to be about mental toughness and handling pressure when the Test starts on Thursday."