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Starc, named player of the tournament at this year's World Cup, played through the pain in Australia's 169-run loss after straining his right ankle, finishing with seven wickets and batting in the fourth and final innings.
Team physio Alex Kountouris said the ankle had "improved significantly", while Clarke was also hopeful the left-armer could recover in time for the second test starting on Thursday.
"Starcy obviously had some ankle pain and had an injection to bowl in the second innings, but the positive again was the fact that he was able to bowl and do what was needed for the team," Clarke told Melbourne radio station Triple M.
"He batted as well in the second innings and ran between the wickets fine.
"Spending a bit of time with the Australian physio today, he's confident that, although it's a short turnaround before the second test, there's been a lot of positives come out of the last couple of days since he was cortisoned, and things are going really well there."
Should Starc fail to prove his fitness, Australia have Peter Siddle or Pat Cummins in reserve.
England named an unchanged 13-man squad for Lord's.
