England lead this five-match series 1-0 after their thrilling 31-run win at Edgbaston last weekend.
Both Moeen Ali and James Porter, who were released for their respective county teams after not being included in the playing XI at Edgbaston, have been retained.
"I've played with him since we were 14 or 15 - as soon as he came over from Zimbabwe - and we've gone through the same path".
According to ECB, Surrey middle-order batsman Ollie Pope replaces Middlesex's Dawid Malan and Warwickshire seamer Chris Woakes returns to the squad in place of Durham's Ben Stokes, who faces a court hearing this coming week in Bristol.
Twenty-year-old Pope, who replaces Dawid Malan, is now the second highest run-scorer in County Championship Division One, with 684 runs at an average of 85.50.
They will, however, make at least two changes to their side.
So young is Pope that he lists Cook and England skipper Joe Root, two men he will be sharing a dressing-room with at Lord's this week, as childhood heroes.
Ollie Pope has made an exceptional start to his first class career.
Malan has paid for his poor form of late, and Bayliss admits the low-scoring nature of the titanic match in Birmingham has put batsmen under the microscope.
"I've had a great season so far and I am feeling confident enough to go out there and perform well".
"Against South Africa past year, there was quite a lot of spin that came into play and when we played India here previously, spin came into it in the second innings. Not a massive difference but just got to look a bit closer I think".
Knocks of 149 and 51 in India's defeat to England in the first Test at Edgbaston have taken him above the former Australia captain, still banned from worldwide cricket over his role in the ball-tampering scandal on the tour of South Africa in March.
Whatever style of bowling he is asked to confront first-up, Pope said that he doesn't suffer "massively" from nerves and this week he will practise his customary use of visualisation as a technique to overcome them. "I think that every time you turn up here, the surface can be quite different". When I bat at 6 I can be in in the 10th over or for the second new ball.
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