The tourists took a five-match one-day series 3-2 with a six-wicket win at Edgbaston on Tuesday -- a result overshadowed by the controversy surrounding the run out backing up or 'Mankading' of England's Jos Buttler.
"After the one-day series loss, they will be coming hard, we know about that, we expect that and we are ready for that," said Jayawardene, a Test specialist gloveman, after the third day of Sri Lanka's four-day tour match against Northamptonshire at Northampton on Saturday was washed out without a ball bowled.
This tour sees Sri Lanka facing England in two Tests, with the first at Lord's next week before the teams meet again at Yorkshire's Headingley headquarters in Leeds.
Sri Lanka have won Tests in England, in 1998 and 2006, but never a series involving more than one match.
During their last Test series in England three years ago, they lost a three-match campaign 1-0 with two draws after a dramatic collapse in Cardiff saw Sri Lanka skittled out for 82.
However, the