The 32-year-old injured 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," head coach Graham Ford told BBC Sport.
Sri Lanka's latest tour match, against Leicestershire at Grace Road, ended in a draw on Sunday.
Prasad took five wickets to set up Sri Lanka's series-clinching victory over England at Headingley two years ago.
"Prasad has had a few tests and the specialist came to see him today," Ford 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."
The tourists enjoyed a better day with the bat on the final day of their three-day match at Grace Road.
Leicestershire batsman Tom Wells hit a career-best unbeaten 87 but was stopped short of completing his ton with the home side unaware they were required to declare after 100 overs.
Sri Lanka opener Dimuth Karunaratne made an untroubled century on the flat pitch and Lahiru Thirimanne hit 40.
"As we have gone along players have started to find their feet and find a bit of form," Ford added. "Every day, there is someone new who is starting to look like he's 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.
"Over the next day or two we'll decide the team but most of the places are starting to sort themselves out."
Sri Lanka bowler Dhammika Prasad has been ruled out of the first Test against England at Headingley, which starts on Thursday.