In his absence, England suffered defeats to Sri Lanka and Australia to jeopardise their chances of qualifying for the knockout stage.
England now need to win at least one and possibly both of their remaining games - against India and New Zealand (July 3) - to progress to the semi-finals.
"Jason Roy is making good progress from his hamstring injury. He is being assessed every day. Yesterday he batted in the nets and was also running shuttles on the outfield," the England and Wales Cricket Board said in a statement.
"A decision on whether he'll be fit to resume against India will be made when we train on Friday and Saturday at Edgbaston."
Roy was in rampaging form before his injury, scoring 215 runs at 71.66 from just four matches. He smashed the second-highest individual score of the World Cup so far with his 153 against Bangladesh.
James Vince was roped in as his replacement but the opener has so far scored 26, 14 and 0 in his last three innings.