Cook was unable to bat in his usual opening position on Saturday after being struck below the left knee while fielding close in earlier in the day.
But as rain prevented play starting as scheduled at 11:00 am local time (100 GMT) Sunday, Cook made his way across the main ground for a fitness test batting in the nets at the Nursery End of Lord's.
An England statement said Cook's injury had "significantly improved overnight" and team officials remain optimistic the 31-year-old left-hander, who made 85 in his side's first innings at Lord's, will be able to bat again in this match.
They then decided to bring lunch forward 30 minutes to 12:30pm (1130 GMT) in the hope of re-starting play in the afternoon session.
England, bidding for a whitewash at 2-0 up in the three-match series, are 109 for four in their second innings -- a lead of 237 runs.
Alex Hales is 41 not out and nightwatchman Steven Finn six not out.