The ECB said the 23-year-old had practised with the full squad over the past two days.
England have a final practice session at the MA Chidambaram Stadium in Chennai on Thursday ahead of the first Test starting in the south Indian city on Friday.
“… The England medical team are satisfied that he (Pope) is available for selection,” the ECB said in a statement.
Pope had initially been ruled out of action for up to four months after he sustained the injury in August while saving a boundary in the third Test against Pakistan in Southampton.
He had previously missed three months of cricket after dislocating the same shoulder playing for Surrey.
The first two matches of the four-Test series will be played in Chennai before the teams travel to Ahmedabad for the last two games. India beat the touring England side 4-0 in 2016-17 before losing 4-1 in England in 2018.