Atapattu, 35, was forced out of the recent home Pakistan series with a painful lower back injury that left him with numbness in his legs.
Sri Lanka's selectors are meeting next week to finalise their selections and Atapattu will be given until April 16, three days before the squad departs, to prove his fitness.
''Marvan's injury is improving slowly and the disc bulge in his lower back is getting smaller, reducing the pressure on the nerves,'' physiotherapist Tommy Simsek told reporters.
''But there is no guarantee and if it does not respond fully to treatment then he may need surgery,'' he added.
''We should get a better picture of his condition within the next couple of weeks.'' Sri Lanka's veteran swing bowler Chaminda Vaas, who also missed the Pakistan series because of a side strain, will return to the squad.
Vaas is now operating off a full run-up in the practice nets.
Sri Lanka are also hoping that left-arm fast bowler Ruchira Perera recovers from a hamstring pull.
''Ruchira needs to rehabilitate the injury very thoroughly,'' Simsek said.
''He is getting better but is not 100 per cent fit, he might have to pass a fitness Test by early next week.'' Sri Lanka will play three Tests and five-one internationals during the 11-week tour of England.