Broad had a scan on his left heel two days ago and the injury will see him miss the two in the shortest international format next week. Broad is due to return to India next month, for the final two matches of a one-day international series.
Morgan had previously captained England against Ireland in a one-day international last year.
Twenty20 captain Broad will also miss two matches against India in the shortest international format next week.
"Stuart Broad has been ruled out of the upcoming two-match Twenty20 series against India with a bruised left heel," an ECB statement read.
"Broad will return to the UK immediately to begin a period of rest and rehabilitation."
Broad is due to return to India next month, for the final two matches of a one-day international series.
Twenty-two-year-old Harris is currently in India with the England Performance Programme squad in Pune.
Finn was already to be rested for next week's Twenty20s, but it is hoped he will still be available for the ODI series despite the disc strain in his back.